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
AbstractโIn 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 TermsโErgodic 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 (๐) transmits a signal ๐ 0 which has an average power of๐
๐= E
(โฃ๐ 0โฃ2) to the destination (๐ท) via a single half-duplexrelay (๐ ) 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 ยฉ2010 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 ๐ โ ๐ 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, ๐ผ (interferer) and the relay. It is assumedthat the interferer is transmitting2 a signal ๐ ๐ which has anaverage power of ๐๐ผ
= E(โฃ๐ ๐โฃ2). Furthermore, we assume
that the interference is negligible at the destination and theRโ D link is corrupted only by the noise. Under this scenario,we have
๐ฆ๐ = โ
๐๐ ๐ 0 + โ
๐ผ๐ ๐ ๐ + ๐
๐ . (1)
๐ฆ๐ท= ๐บโ
๐ ๐ทโ
๐๐ ๐ 0 +๐บโ
๐ ๐ทโ
๐ผ๐ ๐ ๐ +๐บโ
๐ ๐ท๐
๐ + ๐
๐ท(2)
where ๐ฆ๐
and ๐ฆ๐ท
represent the received signals at the relayand the destination respectively. Moreover, โ๐๐
, โ๐ ๐ท
and โ๐ผ๐
denote the channel coefficients of the ๐ โ ๐ , ๐ โ ๐ท and๐ผ โ ๐ links respectively. In order to incorporate the effectof multipath fading, โฃโ๐๐
โฃ, โฃโ๐ ๐ท
โฃ and โฃโ๐ผ๐ โฃ are assumed to
be modeled as independent Rayleigh random variables (RVโs)with E
(โฃโ๐๐ โฃ2) = ๐2
1 , E(โฃโ
๐ ๐ทโฃ2) = ๐2
2 , E(โฃโ
๐ผ๐ โฃ2) = ๐2
3 . Inaddition, slowly-varying flat-fading channels are considered inthis paper. Similarly, ๐
๐ and ๐
๐ทare assumed to be mutually
independent and represent the additive white Gaussian noise(AWGN) with a zero-mean and a variance of ๐0 at the inputsof the relay and the destination respectively. Lastly, ๐บ denotesthe amplification gain of the relay and in case of fixed-gainrelay it is given as [2, eq. (13)]
๐บ2 = E
(๐
๐
๐๐โฃโ
๐๐ โฃ2 + ๐
๐ผโฃโ
๐ผ๐ โฃ2 +๐0
)(3)
where ๐๐
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 ๐ผ employs Gaussiancodebook i.e., conditioned on the channel coefficient โ๐ผ๐ , the interferenceterm is Gaussian distributed.
Using (1) and (2), the end-to-end SINR at the destinationcan be written as
๐พ๐ท=
๐บ2โฃโ๐๐
โฃ2โฃโ๐ ๐ท
โฃ2๐๐
๐บ2โฃโ๐ผ๐ โฃ2โฃโ
๐ ๐ทโฃ2๐
๐ผ+๐บ2โฃโ
๐ ๐ทโฃ2๐0 +๐0
. (4)
After re-adjusting the above expression, we can write
๐พ๐ท=
๐พ1๐พ2๐พ2(๐พ3 + 1) + ๐ถ
(5)
where ๐พ1 โ โฃโ๐๐
โฃ2 ๐๐
๐0, ๐พ2 โ โฃโ
๐ ๐ทโฃ2 ๐๐
๐0, ๐พ3 โ โฃโ
๐ผ๐ โฃ2 ๐๐ผ
๐0and
๐ถ โ ๐๐
๐บ2๐0. Note that ๐พ1, ๐พ2 and ๐พ3 are independent exponen-
tial RVโs with E (๐พ1) = ๐พ1 = ๐21๐
๐
๐0, E (๐พ2) = ๐พ2 = ๐2
2๐
๐
๐0
and E (๐พ3) = ๐พ3 = ๐23๐
๐ผ
๐0.
Now using (3), ๐ถ can be expressed as
๐ถ =
[E
(1
๐พ1 + ๐พ3 + 1
)]โ1
. (6)
It is worth mentioning here that (5) represents the generalizedscenario of [2] i.e., it reduces to [2, eq (6) ] when ๐พ3 = 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 ๐ถ using (5).Furthermore, the closed-form expressions for evaluating ๐ถ arepresented in the Appendix I.
Outage capacity (๐ถ๐๐ข๐ก) is defined as the probability that thechannel capacity falls below a predetermined threshold ๐ ๐ .Mathematically3, it can be written as
๐ถ๐๐ข๐ก = ๐
(๐ต
2log2 (1 + ๐พ
๐ท) โค ๐ ๐
)= ๐
(๐พ
๐ทโค 4
๐ ๐๐ต โ 1
)(7)
where ๐ (.) denotes probability and ๐ต represents the channelbandwidth.
Substituting (5) into (7), we have
๐ถ๐๐ข๐ก = ๐
(๐พ1 โค ๐พ2(๐พ3 + 1)๐พ๐กโ + ๐ถ๐พ๐กโ
๐พ2
)(8)
where ๐พ๐กโ โ 4๐ ๐๐ต โ 1.
As mentioned above since ๐พ1, ๐พ2 and ๐พ3 follow exponentialdistributions, therefore we can write
๐ถ๐๐ข๐ก =1
๐พ3
โซ โ
0
๐ผ1 (๐พ3)exp
(โ๐พ3๐พ3
)๐๐พ3 (9)
where ๐ผ1 (๐พ3) โ 1๐พ2
โซ โ
0
[1โ exp
(โ(๐พ3+1)๐พ๐กโ
๐พ1โ ๐ถ๐พ๐กโ
๐พ1๐พ2
)]รexp
(โ ๐พ2
๐พ2
)๐๐พ2.
After some manipulations, the integral ๐ผ1 (๐พ3) can be ex-pressed as
3The 12
in (7) is due to the fact that the relay operates in the half-duplexmode.
357
๐ผ1 (๐พ3) = 1โ 1
๐พ2
exp
(โ (๐พ3 + 1) ๐พ๐กโ๐พ1
)โซ โ
0
exp
(โ๐ถ๐พ๐กโ๐พ1๐พ2
)รexp
(โ ๐พ2
๐พ2
)๐๐พ2.
(10)Now using [8, eq. (3.471.9)], the above integral can be
written in a closed-form as
๐ผ1 (๐พ3) = 1โ2
โ๐ถ๐พ๐กโ๐พ1๐พ2
exp
(โ (๐พ3 + 1) ๐พ๐กโ๐พ1
)๐พ1
(2
โ๐ถ๐พ๐กโ๐พ1๐พ2
)(11)
where ๐พ1 (.) 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
๐ถ๐๐ข๐ก = 1โ 2๐ฟ
๐ฟ+ ๐พ๐กโ
โ๐ถ๐พ๐กโ๐พ1๐พ2
exp
(โ๐พ๐กโ๐พ1
)๐พ1
(2
โ๐ถ๐พ๐กโ๐พ1๐พ2
)(12)
where ๐ฟ โ ๐พ1
๐พ3and 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 ๐พ2 โ โ),(12) reduces to a simpler expression as follows
๐ถ๐๐ข๐ก = 1โ ๐ฟ
๐ฟ+ ๐พ๐กโexp
(โ๐พ๐กโ๐พ1
). (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
๐พ๐ก =๐พ1๐พ2
๐พ2๐พ3 + ๐ถ(14)
where in this case ๐ถ can easily be obtained from [2] as
๐ถ =
[E
(1
๐พ1 + ๐พ3
)]โ1
. (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) ๐พ1 = ๐พ3 (๐ฟ = 1)and ii) ๐พ1 โ= ๐พ3 (๐ฟ โ= 1).
A. Case I: ๐พ1 = ๐พ3 = ๐พ (๐ฟ = 1)
Since the ergodic capacity (per unit channel bandwidth),(๐ถ๐ธ), of the dual-hop network (with channel informationavailable only to the receiving terminals) is given as
๐ถ๐ธ =1
2E [log2 (1 + ๐พ๐ก)] . (16)
Therefore substituting (14) into (16) and after re-adjusting wehave
๐ถ๐ธ =1
2ln(2)
โกโขโฃE (ln (๐ง๐พ2 + ๐ถ))๏ธธ ๏ธท๏ธท ๏ธธ
โ๐ด
โE (ln (๐พ2๐พ3 + ๐ถ))๏ธธ ๏ธท๏ธท ๏ธธโ๐ต
โคโฅโฆ (17)
where ln (.) denotes the natural logarithm and ๐ง โ ๐พ1 + ๐พ3with its probability density function (PDF) given by [10]
๐๐ง(๐ง) =๐ง
๐พ2 exp
(โ ๐ง
๐พ
). (18)
Now solving ๐ด in (17), we can have
๐ด =
โซ โ
0
[1
๐พ2
โซ โ
0
ln (๐ง๐พ2 + ๐ถ)exp
(โ๐พ2๐พ2
)๐๐พ2
]๐๐ง(๐ง)๐๐ง.
(19)The above integral inside the square brackets can be further
solved using [8, eq. (4.337.1)] and the identity โ๐ธi (โ๐ฅ) =ฮ (0, ๐ฅ) for ๐ฅ > 0
๐ด =
โซ โ
0
[ln(๐ถ) + exp
(๐ถ
๐พ2๐ง
)ฮ
(0,
๐ถ
๐พ2๐ง
)]๐๐ง(๐ง)๐๐ง (20)
where above ๐ธ๐ (.)and ฮ (, ., ) 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
๐ด = ln(๐ถ) +32๐ถ
๐พ ๐พ2
Sโ3,2
(2
โ๐ถ
๐พ ๐พ2
)(21)
where S๐,๐ฃ (.) denotes the Lommel function and is availableas a built-in function in Maple.
Similarly (like the derivation of ๐ด), ๐ต can be written in aclosed-form as
๐ต = ln(๐ถ) + 8
โ๐ถ
๐พ ๐พ2
Sโ2,1
(2
โ๐ถ
๐พ ๐พ2
). (22)
Lastly, substituting (21) and (22) into (17) we get (25).
B. Case II: ๐พ1 โ= ๐พ3 (๐ฟ โ= 1)
In this case, the PDF of ๐ง (which is a sum of two non-identical exponential random variables) is given as
๐๐ง(๐ง) =1
(๐พ1 โ ๐พ3)
[exp
(โ๐ง
๐พ1
)โ exp
(โ๐ง
๐พ3
)]. (23)
Now substituting (23) into (20) and following the same deriva-tion as we did before for (21), we have a closed-form for ๐ดas
358
๐ด= ln(๐ถ)+8
(๐พ1 โ ๐พ3)
โ๐ถ
๐พ2
[โ๐พ1 Sโ2,1
(2
โ๐ถ
๐พ1 ๐พ2
)
โโ๐พ3 Sโ2,1
(2
โ๐ถ
๐พ2 ๐พ3
)].
(24)Note that the expression for ๐ต remains exactly the same as in(22) except ๐พ is replaced by ๐พ3. 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 (๐ฟ). As expected, outagecapacity decreases with decreasing the interference level (i.e.,increasing ๐ฟ). However this trend is more pronounced at lowto moderate values of ๐พ. Moreover, at sufficiently high valuesof ๐พ and for a fixed value of ๐ฟ, the outage capacity remainsalmost constant. A similar trend is also observed from Fig. 3in which the outage capacity is plotted for different values of๐พ2. In this figure, it is shown that for fixed values of ๐ฟ and๐พ3, the outage capacity remains almost constant at asymptoticvalues of ๐พ2. 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 of๐ฟ. Again it is evident from Fig. 4 that the ergodic capacityincreases with increasing ๐ฟ and for a set of any two values of๐ฟ, the corresponding gap is more at moderate to high values of๐พ. 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 ๐ถ 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 gain๐ถ.
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
ฮณ (dB)
Outa
ge C
apac
ity (
C out)
L=1.0L=10L=20Simulations
Figure 2. Plot of the outage capacity for different values of ๐พ using (12)with ๐พ๐กโ = 1, ๐พ1 = ๐พ2 = ๐พ3 = ๐พ for ๐ฟ = 1 and ๐พ2 = ๐พ3 = ๐พ = ๐๐พ1 for
๐ฟ = ๐ with ๐ =
{1
10,1
20
}.
10 20 30 40 50 60 70 80 90 10010-2
10-1
ฮณ2 (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 ๐พ2 using (12)and (13) with ๐พ3 = 10 dB, ๐พ1 = 20 dB and ๐ฟ = 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 ๐ถ which is required in (12). Similar to SectionIV, we consider two scenarios here, i.e., i) ๐พ1 = ๐พ3 (๐ฟ = 1)and ii) ๐พ1 โ= ๐พ3 (๐ฟ โ= 1).
Case I: ๐พ1 = ๐พ3 = ๐พ
Using (6) and [12, eq. (5b)], we can write
๐ถ =
[โซ โ
0
exp (โ๐ )
(1 + ๐ ๐พ)2 ๐๐
]โ1
. (27)
359
๐ถ๐ธ =4
ln(2)
โ๐ถ
๐พ ๐พ2
[4
โ๐ถ
๐พ ๐พ2
Sโ3,2
(2
โ๐ถ
๐พ ๐พ2
)โ Sโ2,1
(2
โ๐ถ
๐พ ๐พ2
)]. (Case I) (25)
๐ถ๐ธ =4
ln(2) (๐พ1 โ ๐พ3)
โ๐ถ
๐พ2
[โ๐พ1 Sโ2,1
(2
โ๐ถ
๐พ1 ๐พ2
)โ ๐พ1โ
๐พ3
Sโ2,1
(2
โ๐ถ
๐พ2 ๐พ3
)]. (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
ฮณ (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). ๐พ1 = ๐พ2 = ๐พ3 =๐พ for ๐ฟ = 1 and ๐พ1 = ๐พ2 = ๐พ = ๐๐พ3 for ๐ฟ = ๐ with ๐ = {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
๐ถ =
[1
๐พexp
(1
2๐พ
)Wโ1,โ 1
2
(1
๐พ
)]โ1
(28)
where W (.) is the Whittaker function defined in [8].
Case II: ๐พ1 โ= ๐พ3
Using (6) and (23), we can write
๐ถ =
[โซ โ
0
1
๐ง + 1๐๐ง(๐ง)๐๐ง
]โ1
(29)
where as above ๐ง โ ๐พ1 + ๐พ3.Therefore applying (23), [8, eq. (3.352.4)] and using the
identityโ๐ธ๐ (โ๐ฅ) = ๐ธ1 (๐ฅ) , ๐ถ can be written in a closed-form as
๐ถ =๐พ1 โ ๐พ3
exp(
1๐พ1
)๐ธ1
(1๐พ1
)โ exp
(1๐พ3
)๐ธ1
(1๐พ3
) (30)
where ๐ธ1 (.) is the exponential integral defined in [8].
APPENDIX II
Here we will derive the closed-form expression of ๐ถ forthe interference-limited scenario and these expressions arerequired in the Section IV to evaluate (25) and (26).
From (15), we have
๐ถ=
[1
๐พ1
1
๐พ3
โซ โ
0
โซ โ
0
1
(๐พ1 + ๐พ3)exp
(โ๐พ1๐พ1
)exp
(โ๐พ3๐พ3
)๐๐พ1๐๐พ3
]โ1
.
(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
๐ถ =
[1
๐พ1
2๐น1
(1, 1; 2; 1โ ๐พ3
๐พ1
)]โ1
(32)
where 2๐น1 (., .; .; .) is the Gaussโ hypergeometric functiondefined in [9, eq. (15.1.1)]. Furthermore for a special caseof ๐พ1 = ๐พ3 = ๐พ, we have 2๐น1 (1, 1; 2; 0) = 1, therefore theabove expression reduces to
๐ถ = ๐พ. (33)
Moreover, for ๐พ1 โ= ๐พ3 we can use the identity [9, eq. (15.1.3)]and so (32) simplifies to
๐ถ =๐พ1 โ ๐พ3
ln (๐ฟ). (34)
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