Outage and Ergodic Capacity Expressions forFixed-Gain Relay Networks in the Presence of
InterferenceOmer Waqar1, Mounir Ghogho1,2 and Des McLernon1
1. School of Electronic & Electrical Engineering, University of Leeds, Leeds, UK.2. International University of Rabat, Morocco.
email:{elow, m.ghogho, d.c.mclernon}@leeds.ac.uk
AbstractβIn this paper, we analyze the outage and ergodiccapacities of a dual-hop cooperative system equipped with a singlefixed-gain (also known as semi-blind) relay whilst considering aninterference at the relay terminal. We derive the expression forend-to-end signal-to-interference and noise ratio (SINR) of thesystem under consideration. Using this expression, we then derivenew exact closed-form expressions for the outage and ergodiccapacities assuming flat Rayleigh fading conditions. Furthermore,new analytical closed-form expressions are derived for the gain ofrelay which are then used in the numerical simulations. Finally,we also provide the Monte Carlo simulations and show theirexact match with our derived closed-form expressions.
Index TermsβErgodic capacity, fixed-gain relay, interference,Lommel functions and outage capacity .
I. INTRODUCTION
DUAL-HOP relaying is usually used in bent-pipe satelliteswhere the primary function of the spacecraft is to relay
the uplink carrier into a downlink [1]. Depending upon thenature of the relays, dual-hop networks are classified aseither (i) Amplify-and-forward (non-regenerative) systems or(ii) Decode-and-forward (regenerative) systems. Amplify-and-forward is a relatively simple protocol in which relays onlyamplify the source signal and forward it to the destinationwithout performing any decoding. Non-Regenerative networksare further categorized into two types: Variable-gain relaynetworks and Fixed-gain relay networks. In variable-gainrelay networks, a relay simply amplifies the signal using theinstantaneous Channel State Information (CSI) of the previoushop, whereas fixed-gain relays do not require instantaneousCSI, but only need a knowledge of the average fading powerof the previous hop. Therefore from a practical point of view,fixed-gain relay networks are relatively less complex and easyto deploy when compared to variable-gain relay networks [2].Realizing the importance and advantages of fixed-gain relaynetworks, in this paper we consider a dual-hop transmissionsystem equipped with a single fixed-gain relay and develop anew analytical framework for its outage and ergodic capacitiesassuming Rayleigh fading channels and interference at therelay terminal.
During the last few years, substantial research work hasbeen carried out on the performance analysis of fixed-gainrelay networks under various fading environments (e.g., see
[3]-[5] and the references therein). However in all these con-tributions an ideal configuration has been considered i.e., nointerference has been considered at the relay and it is assumedthat the source signal arriving at the relay is perturbed only bythe noise. Moreover, recently the case of interference-limiteddestination has been considered in [6] and a closed-form ex-pression for the outage probability has been derived. However,in this work it is assumed that the interference is present atthe destination only and the relay is perturbed only by thenoise. In this paper, we try to remove the abovementionedlimitations and consider a more realistic scenario in which thesignal arriving at the fixed-gain relay is corrupted both by thenoise1 and the interference. This scenario is already discussedin [7] for the variable-gain relays but in a different context.
With respect to the previously reported results, our contribu-tions in this paper are threefold: i) We derive an expression forthe end-to-end SINR which appears to be the generalized caseof [2]; ii) Based on this SINR expression we then derive thenew exact closed-form expressions for the outage and ergodiccapacities over the Rayleigh fading channels and iii) Closed-form expressions for the gain of the relay have been derivedwhich are then used in the numerical simulations.
The remainder of the paper is organized as follows. InSection II, we describe the system and channel models underconsideration and also derive the expression for the end-to-end SINR. Based on the results of Section II, closed-formexpressions for the outage capacity are derived in Section III.Similarly, using the result of Section II, we derive the exactclosed-form expressions for the ergodic capacity in the SectionIV. Moreover, in Section V, we give some numerical resultsand show their validity by comparing them with the resultsobtained through Monte Carlo simulations. Finally, concludingremarks are presented in Section VI.
II. SYSTEM AND CHANNEL MODELS
We consider a dual-hop wireless relaying system in which asource (π) transmits a signal π 0 which has an average power ofπ
π= E
(β£π 0β£2) to the destination (π·) via a single half-duplexrelay (π ) as shown in Fig. 1. We assume that in addition to
1For mathematical tractability, we consider an interference-limited relayterminal (i.e., noise is neglected) during the derivation of the exact closed-form expressions for the ergodic capacity.
978-1-4244-7006-8/10/$26.00 Β©2010 IEEE 356
Figure 1. Illustration of the system and channel models showing the inter-cluster interference.
the noise, the inter-cluster interference is present at the relaywhich degrades the π β π link (see [7] for more detailsabout the inter-cluster interference). As shown in the Fig. 1,this interference can be treated as a direct path between theneighbouring source, πΌ (interferer) and the relay. It is assumedthat the interferer is transmitting2 a signal π π which has anaverage power of ππΌ
= E(β£π πβ£2). Furthermore, we assume
that the interference is negligible at the destination and theRβ D link is corrupted only by the noise. Under this scenario,we have
π¦π = β
ππ π 0 + β
πΌπ π π + π
π . (1)
π¦π·= πΊβ
π π·β
ππ π 0 +πΊβ
π π·β
πΌπ π π +πΊβ
π π·π
π + π
π·(2)
where π¦π
and π¦π·
represent the received signals at the relayand the destination respectively. Moreover, βππ
, βπ π·
and βπΌπ
denote the channel coefficients of the π β π , π β π· andπΌ β π links respectively. In order to incorporate the effectof multipath fading, β£βππ
β£, β£βπ π·
β£ and β£βπΌπ β£ are assumed to
be modeled as independent Rayleigh random variables (RVβs)with E
(β£βππ β£2) = π2
1 , E(β£β
π π·β£2) = π2
2 , E(β£β
πΌπ β£2) = π2
3 . Inaddition, slowly-varying flat-fading channels are considered inthis paper. Similarly, π
π and π
π·are assumed to be mutually
independent and represent the additive white Gaussian noise(AWGN) with a zero-mean and a variance of π0 at the inputsof the relay and the destination respectively. Lastly, πΊ denotesthe amplification gain of the relay and in case of fixed-gainrelay it is given as [2, eq. (13)]
πΊ2 = E
(π
π
ππβ£β
ππ β£2 + π
πΌβ£β
πΌπ β£2 +π0
)(3)
where ππ
is the power of the transmitted signal at the outputof the relay. Note that E (.) denotes the statistical expectationoperator throughout this paper.
2In this paper, we assume that the interfering source πΌ employs Gaussiancodebook i.e., conditioned on the channel coefficient βπΌπ , the interferenceterm is Gaussian distributed.
Using (1) and (2), the end-to-end SINR at the destinationcan be written as
πΎπ·=
πΊ2β£βππ
β£2β£βπ π·
β£2ππ
πΊ2β£βπΌπ β£2β£β
π π·β£2π
πΌ+πΊ2β£β
π π·β£2π0 +π0
. (4)
After re-adjusting the above expression, we can write
πΎπ·=
πΎ1πΎ2πΎ2(πΎ3 + 1) + πΆ
(5)
where πΎ1 β β£βππ
β£2 ππ
π0, πΎ2 β β£β
π π·β£2 ππ
π0, πΎ3 β β£β
πΌπ β£2 ππΌ
π0and
πΆ β ππ
πΊ2π0. Note that πΎ1, πΎ2 and πΎ3 are independent exponen-
tial RVβs with E (πΎ1) = πΎ1 = π21π
π
π0, E (πΎ2) = πΎ2 = π2
2π
π
π0
and E (πΎ3) = πΎ3 = π23π
πΌ
π0.
Now using (3), πΆ can be expressed as
πΆ =
[E
(1
πΎ1 + πΎ3 + 1
)]β1
. (6)
It is worth mentioning here that (5) represents the generalizedscenario of [2] i.e., it reduces to [2, eq (6) ] when πΎ3 = 0 (i.e.,no interference at the relay).
III. OUTAGE CAPACITY
In this section we will derive new exact closed-form ex-pressions for the outage capacity in terms of πΆ using (5).Furthermore, the closed-form expressions for evaluating πΆ arepresented in the Appendix I.
Outage capacity (πΆππ’π‘) is defined as the probability that thechannel capacity falls below a predetermined threshold π π .Mathematically3, it can be written as
πΆππ’π‘ = π
(π΅
2log2 (1 + πΎ
π·) β€ π π
)= π
(πΎ
π·β€ 4
π ππ΅ β 1
)(7)
where π (.) denotes probability and π΅ represents the channelbandwidth.
Substituting (5) into (7), we have
πΆππ’π‘ = π
(πΎ1 β€ πΎ2(πΎ3 + 1)πΎπ‘β + πΆπΎπ‘β
πΎ2
)(8)
where πΎπ‘β β 4π ππ΅ β 1.
As mentioned above since πΎ1, πΎ2 and πΎ3 follow exponentialdistributions, therefore we can write
πΆππ’π‘ =1
πΎ3
β« β
0
πΌ1 (πΎ3)exp
(βπΎ3πΎ3
)ππΎ3 (9)
where πΌ1 (πΎ3) β 1πΎ2
β« β
0
[1β exp
(β(πΎ3+1)πΎπ‘β
πΎ1β πΆπΎπ‘β
πΎ1πΎ2
)]Γexp
(β πΎ2
πΎ2
)ππΎ2.
After some manipulations, the integral πΌ1 (πΎ3) can be ex-pressed as
3The 12
in (7) is due to the fact that the relay operates in the half-duplexmode.
357
πΌ1 (πΎ3) = 1β 1
πΎ2
exp
(β (πΎ3 + 1) πΎπ‘βπΎ1
)β« β
0
exp
(βπΆπΎπ‘βπΎ1πΎ2
)Γexp
(β πΎ2
πΎ2
)ππΎ2.
(10)Now using [8, eq. (3.471.9)], the above integral can be
written in a closed-form as
πΌ1 (πΎ3) = 1β2
βπΆπΎπ‘βπΎ1πΎ2
exp
(β (πΎ3 + 1) πΎπ‘βπΎ1
)πΎ1
(2
βπΆπΎπ‘βπΎ1πΎ2
)(11)
where πΎ1 (.) is the first-order modified Bessel function of thesecond kind and is defined in [9, eq. (9.6.22)].
Finally, substituting (11) into (9) and using the result of [8,eq. (3.310)], we get
πΆππ’π‘ = 1β 2πΏ
πΏ+ πΎπ‘β
βπΆπΎπ‘βπΎ1πΎ2
exp
(βπΎπ‘βπΎ1
)πΎ1
(2
βπΆπΎπ‘βπΎ1πΎ2
)(12)
where πΏ β πΎ1
πΎ3and it quantifies the level of interference at the
fixed-gain relay. To the best of our knowledge, (12) is a newclosed-form expression which exactly evaluates the outagecapacity of the system under consideration. Moreover, weshow that in the asymptotic regime (i.e., in case of πΎ2 β β),(12) reduces to a simpler expression as follows
πΆππ’π‘ = 1β πΏ
πΏ+ πΎπ‘βexp
(βπΎπ‘βπΎ1
). (13)
IV. ERGODIC CAPACITY
In this section, we will derive the exact closed-form expres-sions for the ergodic capacity. In order to make the expressionsmathematical tractable, we assume that the noise is negligibleat the input of the fixed-gain relay i.e., interference-limitedenvironment. In this particular scenario, the expression for theend-to-end SINR in (5) now becomes
πΎπ‘ =πΎ1πΎ2
πΎ2πΎ3 + πΆ(14)
where in this case πΆ can easily be obtained from [2] as
πΆ =
[E
(1
πΎ1 + πΎ3
)]β1
. (15)
Note that the closed-form expressions of (15) have beenderived in the Appendix II. However, next we will derivethe exact closed-form expressions for the ergodic capacityconsidering the two cases separately i.e., i) πΎ1 = πΎ3 (πΏ = 1)and ii) πΎ1 β= πΎ3 (πΏ β= 1).
A. Case I: πΎ1 = πΎ3 = πΎ (πΏ = 1)
Since the ergodic capacity (per unit channel bandwidth),(πΆπΈ), of the dual-hop network (with channel informationavailable only to the receiving terminals) is given as
πΆπΈ =1
2E [log2 (1 + πΎπ‘)] . (16)
Therefore substituting (14) into (16) and after re-adjusting wehave
πΆπΈ =1
2ln(2)
β‘β’β£E (ln (π§πΎ2 + πΆ))οΈΈ οΈ·οΈ· οΈΈ
βπ΄
βE (ln (πΎ2πΎ3 + πΆ))οΈΈ οΈ·οΈ· οΈΈβπ΅
β€β₯β¦ (17)
where ln (.) denotes the natural logarithm and π§ β πΎ1 + πΎ3with its probability density function (PDF) given by [10]
ππ§(π§) =π§
πΎ2 exp
(β π§
πΎ
). (18)
Now solving π΄ in (17), we can have
π΄ =
β« β
0
[1
πΎ2
β« β
0
ln (π§πΎ2 + πΆ)exp
(βπΎ2πΎ2
)ππΎ2
]ππ§(π§)ππ§.
(19)The above integral inside the square brackets can be further
solved using [8, eq. (4.337.1)] and the identity βπΈi (βπ₯) =Ξ (0, π₯) for π₯ > 0
π΄ =
β« β
0
[ln(πΆ) + exp
(πΆ
πΎ2π§
)Ξ
(0,
πΆ
πΎ2π§
)]ππ§(π§)ππ§ (20)
where above πΈπ (.)and Ξ (, ., ) represent the exponential inte-gral and the upper incomplete Gamma function respectivelyand are defined in [8].
Finally, using (18) and [11, eq. (3.10.3.3)], (20) can beexpressed in closed-form as
π΄ = ln(πΆ) +32πΆ
πΎ πΎ2
Sβ3,2
(2
βπΆ
πΎ πΎ2
)(21)
where Sπ,π£ (.) denotes the Lommel function and is availableas a built-in function in Maple.
Similarly (like the derivation of π΄), π΅ can be written in aclosed-form as
π΅ = ln(πΆ) + 8
βπΆ
πΎ πΎ2
Sβ2,1
(2
βπΆ
πΎ πΎ2
). (22)
Lastly, substituting (21) and (22) into (17) we get (25).
B. Case II: πΎ1 β= πΎ3 (πΏ β= 1)
In this case, the PDF of π§ (which is a sum of two non-identical exponential random variables) is given as
ππ§(π§) =1
(πΎ1 β πΎ3)
[exp
(βπ§
πΎ1
)β exp
(βπ§
πΎ3
)]. (23)
Now substituting (23) into (20) and following the same deriva-tion as we did before for (21), we have a closed-form for π΄as
358
π΄= ln(πΆ)+8
(πΎ1 β πΎ3)
βπΆ
πΎ2
[βπΎ1 Sβ2,1
(2
βπΆ
πΎ1 πΎ2
)
ββπΎ3 Sβ2,1
(2
βπΆ
πΎ2 πΎ3
)].
(24)Note that the expression for π΅ remains exactly the same as in(22) except πΎ is replaced by πΎ3. Finally, using (17), (22) and(24), we get a closed-form expression for the ergodic capacitywhich is shown on the top of next page as (26). To the bestof our knowledge, (25) and (26) are new and exactly evaluatethe ergodic capacity of the dual-hop fixed-gain relay networkin the interference-limited scenario (where noise is negligibleat the relay but present at the input of destination).
V. NUMERICAL AND SIMULATION RESULTS
In this section, we discuss and analyze the numerical resultsderived in the Sections III and IV and verify their validityby showing that they match exactly with the Monte Carlosimulations. In Fig. 2, we plot the outage capacity for differentvalues of the interference levels (πΏ). As expected, outagecapacity decreases with decreasing the interference level (i.e.,increasing πΏ). However this trend is more pronounced at lowto moderate values of πΎ. Moreover, at sufficiently high valuesof πΎ and for a fixed value of πΏ, the outage capacity remainsalmost constant. A similar trend is also observed from Fig. 3in which the outage capacity is plotted for different values ofπΎ2. In this figure, it is shown that for fixed values of πΏ andπΎ3, the outage capacity remains almost constant at asymptoticvalues of πΎ2. In addition to this, it is clear from both Fig. 2and Fig. 3 that the analytical results derived in the Section IIIare indistinguishable from the simulation results. Furthermore,we plot the ergodic capacity in Fig. 4 for different values ofπΏ. Again it is evident from Fig. 4 that the ergodic capacityincreases with increasing πΏ and for a set of any two values ofπΏ, the corresponding gap is more at moderate to high values ofπΎ. Moreover, Fig. 4 also verifies the correctness of the closed-form expressions derived in the Section IV. Finally, note thatwe have used the closed-form expressions of πΆ derived in theAppendices I and II in order to evaluate it and then use itfor calculating the ergodic and outage capacities using (12),(25) and (26). Thus Figs 2-4 also verify the correctness of theclosed-form expressions derived for evaluating the relay gainπΆ.
VI. CONCLUSION
In this paper, we first derive the expression for the end-to-end SINR and then based on this result, we derive thenew exact closed-form expressions for the outage and ergodiccapacities of the dual-hop relay network considering the in-terference at the fixed-gain relay and non-identical Rayleighfading channels. Moreover, we also derive the closed-formexpressions for the gain of the relay. Since the closed-formexpressions given in this paper exactly evaluate the outage andergodic capacities of the required system, thus time-consumingMonte Carlo simulations can be avoided. Furthermore, as a
0 5 10 15 20 25
10-1
100
Ξ³ (dB)
Outa
ge C
apac
ity (
C out)
L=1.0L=10L=20Simulations
Figure 2. Plot of the outage capacity for different values of πΎ using (12)with πΎπ‘β = 1, πΎ1 = πΎ2 = πΎ3 = πΎ for πΏ = 1 and πΎ2 = πΎ3 = πΎ = ππΎ1 for
πΏ = π with π =
{1
10,1
20
}.
10 20 30 40 50 60 70 80 90 10010-2
10-1
Ξ³2 (dB)
Outa
ge C
apac
ity (C
out)
RT/B=1
RT/B=0.5
RT/B=0.25
Asymptotic Simulations
Figure 3. Plot of the outage capacity for different values of πΎ2 using (12)and (13) with πΎ3 = 10 dB, πΎ1 = 20 dB and πΏ = 10.
future work these expressions can also be used for optimizingthe ergodic and outage capacities.
APPENDIX I
Now we present the closed-form expressions for evaluatingthe relay gain πΆ which is required in (12). Similar to SectionIV, we consider two scenarios here, i.e., i) πΎ1 = πΎ3 (πΏ = 1)and ii) πΎ1 β= πΎ3 (πΏ β= 1).
Case I: πΎ1 = πΎ3 = πΎ
Using (6) and [12, eq. (5b)], we can write
πΆ =
[β« β
0
exp (βπ )
(1 + π πΎ)2 ππ
]β1
. (27)
359
πΆπΈ =4
ln(2)
βπΆ
πΎ πΎ2
[4
βπΆ
πΎ πΎ2
Sβ3,2
(2
βπΆ
πΎ πΎ2
)β Sβ2,1
(2
βπΆ
πΎ πΎ2
)]. (Case I) (25)
πΆπΈ =4
ln(2) (πΎ1 β πΎ3)
βπΆ
πΎ2
[βπΎ1 Sβ2,1
(2
βπΆ
πΎ1 πΎ2
)β πΎ1β
πΎ3
Sβ2,1
(2
βπΆ
πΎ2 πΎ3
)]. (Case II) (26)
0 5 10 15 20 250.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Ξ³ (dB)
Ergo
dic C
apac
ity, C
E (bit
s/sec
/Hz)
L=1L=10L=20Simulations
Figure 4. Plot of the ergodic capacity using (25) and (26). πΎ1 = πΎ2 = πΎ3 =πΎ for πΏ = 1 and πΎ1 = πΎ2 = πΎ = ππΎ3 for πΏ = π with π = {10, 20} .
Now using [8, eq. (3.382.3)] and after some manipulation,the integral inside the square brackets can be written in aclosed-form as
πΆ =
[1
πΎexp
(1
2πΎ
)Wβ1,β 1
2
(1
πΎ
)]β1
(28)
where W (.) is the Whittaker function defined in [8].
Case II: πΎ1 β= πΎ3
Using (6) and (23), we can write
πΆ =
[β« β
0
1
π§ + 1ππ§(π§)ππ§
]β1
(29)
where as above π§ β πΎ1 + πΎ3.Therefore applying (23), [8, eq. (3.352.4)] and using the
identityβπΈπ (βπ₯) = πΈ1 (π₯) , πΆ can be written in a closed-form as
πΆ =πΎ1 β πΎ3
exp(
1πΎ1
)πΈ1
(1πΎ1
)β exp
(1πΎ3
)πΈ1
(1πΎ3
) (30)
where πΈ1 (.) is the exponential integral defined in [8].
APPENDIX II
Here we will derive the closed-form expression of πΆ forthe interference-limited scenario and these expressions arerequired in the Section IV to evaluate (25) and (26).
From (15), we have
πΆ=
[1
πΎ1
1
πΎ3
β« β
0
β« β
0
1
(πΎ1 + πΎ3)exp
(βπΎ1πΎ1
)exp
(βπΎ3πΎ3
)ππΎ1ππΎ3
]β1
.
(31)Now using the results [8, eq. (3.352.4)], [8, eq. (6.228.2)] andafter some manipulation, the above expression can be writtenin a closed-form as
πΆ =
[1
πΎ1
2πΉ1
(1, 1; 2; 1β πΎ3
πΎ1
)]β1
(32)
where 2πΉ1 (., .; .; .) is the Gaussβ hypergeometric functiondefined in [9, eq. (15.1.1)]. Furthermore for a special caseof πΎ1 = πΎ3 = πΎ, we have 2πΉ1 (1, 1; 2; 0) = 1, therefore theabove expression reduces to
πΆ = πΎ. (33)
Moreover, for πΎ1 β= πΎ3 we can use the identity [9, eq. (15.1.3)]and so (32) simplifies to
πΆ =πΎ1 β πΎ3
ln (πΏ). (34)
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