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

Abstract—In 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 Terms—Ergodic 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 (𝑆) transmits a signal 𝑠0 which has an average power of𝑃

𝑆= E

(∣𝑠0∣2) to the destination (𝐷) via a single half-duplexrelay (𝑅) 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 ©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 𝑆 → 𝑅 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, 𝐼 (interferer) and the relay. It is assumedthat the interferer is transmitting2 a signal 𝑠𝑖 which has anaverage power of 𝑃𝐼

= E(∣𝑠𝑖∣2). Furthermore, we assume

that the interference is negligible at the destination and theR→ D link is corrupted only by the noise. Under this scenario,we have

𝑦𝑅= ℎ

𝑆𝑅𝑠0 + ℎ

𝐼𝑅𝑠𝑖 + 𝑛

𝑅. (1)

𝑦𝐷= 𝐺ℎ

𝑅𝐷ℎ

𝑆𝑅𝑠0 +𝐺ℎ

𝑅𝐷ℎ

𝐼𝑅𝑠𝑖 +𝐺ℎ

𝑅𝐷𝑛

𝑅+ 𝑛

𝐷(2)

where 𝑦𝑅

and 𝑦𝐷

represent the received signals at the relayand the destination respectively. Moreover, ℎ𝑆𝑅

, ℎ𝑅𝐷

and ℎ𝐼𝑅

denote the channel coefficients of the 𝑆 → 𝑅, 𝑅 → 𝐷 and𝐼 → 𝑅 links respectively. In order to incorporate the effectof multipath fading, ∣ℎ𝑆𝑅

∣, ∣ℎ𝑅𝐷

∣ and ∣ℎ𝐼𝑅∣ are assumed to

be modeled as independent Rayleigh random variables (RV’s)with E

(∣ℎ𝑆𝑅∣2) = 𝜎2

1 , E(∣ℎ

𝑅𝐷∣2) = 𝜎2

2 , E(∣ℎ

𝐼𝑅∣2) = 𝜎2

3 . Inaddition, slowly-varying flat-fading channels are considered inthis paper. Similarly, 𝑛

𝑅and 𝑛

𝐷are assumed to be mutually

independent and represent the additive white Gaussian noise(AWGN) with a zero-mean and a variance of 𝑁0 at the inputsof the relay and the destination respectively. Lastly, 𝐺 denotesthe amplification gain of the relay and in case of fixed-gainrelay it is given as [2, eq. (13)]

𝐺2 = E

(𝑃

𝑅

𝑃𝑆∣ℎ

𝑆𝑅∣2 + 𝑃

𝐼∣ℎ

𝐼𝑅∣2 +𝑁0

)(3)

where 𝑃𝑅

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 𝐼 employs Gaussiancodebook i.e., conditioned on the channel coefficient ℎ𝐼𝑅 , the interferenceterm is Gaussian distributed.

Using (1) and (2), the end-to-end SINR at the destinationcan be written as

𝛾𝐷=

𝐺2∣ℎ𝑆𝑅

∣2∣ℎ𝑅𝐷

∣2𝑃𝑆

𝐺2∣ℎ𝐼𝑅∣2∣ℎ

𝑅𝐷∣2𝑃

𝐼+𝐺2∣ℎ

𝑅𝐷∣2𝑁0 +𝑁0

. (4)

After re-adjusting the above expression, we can write

𝛾𝐷=

𝛾1𝛾2𝛾2(𝛾3 + 1) + 𝐶

(5)

where 𝛾1 ≜ ∣ℎ𝑆𝑅

∣2 𝑃𝑆

𝑁0, 𝛾2 ≜ ∣ℎ

𝑅𝐷∣2 𝑃𝑅

𝑁0, 𝛾3 ≜ ∣ℎ

𝐼𝑅∣2 𝑃𝐼

𝑁0and

𝐶 ≜ 𝑃𝑅

𝐺2𝑁0. Note that 𝛾1, 𝛾2 and 𝛾3 are independent exponen-

tial RV’s with E (𝛾1) = 𝛾1 = 𝜎21𝑃

𝑆

𝑁0, E (𝛾2) = 𝛾2 = 𝜎2

2𝑃

𝑅

𝑁0

and E (𝛾3) = 𝛾3 = 𝜎23𝑃

𝐼

𝑁0.

Now using (3), 𝐶 can be expressed as

𝐶 =

[E

(1

𝛾1 + 𝛾3 + 1

)]−1

. (6)

It is worth mentioning here that (5) represents the generalizedscenario of [2] i.e., it reduces to [2, eq (6) ] when 𝛾3 = 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 𝐶 using (5).Furthermore, the closed-form expressions for evaluating 𝐶 arepresented in the Appendix I.

Outage capacity (𝐶𝑜𝑢𝑡) is defined as the probability that thechannel capacity falls below a predetermined threshold 𝑅𝑇 .Mathematically3, it can be written as

𝐶𝑜𝑢𝑡 = 𝑃

(𝐵

2log2 (1 + 𝛾

𝐷) ≤ 𝑅𝑇

)= 𝑃

(𝛾

𝐷≤ 4

𝑅𝑇𝐵 − 1

)(7)

where 𝑃 (.) denotes probability and 𝐵 represents the channelbandwidth.

Substituting (5) into (7), we have

𝐶𝑜𝑢𝑡 = 𝑃

(𝛾1 ≤ 𝛾2(𝛾3 + 1)𝛾𝑡ℎ + 𝐶𝛾𝑡ℎ

𝛾2

)(8)

where 𝛾𝑡ℎ ≜ 4𝑅𝑇𝐵 − 1.

As mentioned above since 𝛾1, 𝛾2 and 𝛾3 follow exponentialdistributions, therefore we can write

𝐶𝑜𝑢𝑡 =1

𝛾3

∫ ∞

0

𝐼1 (𝛾3)exp

(−𝛾3𝛾3

)𝑑𝛾3 (9)

where 𝐼1 (𝛾3) ≜ 1𝛾2

∫ ∞

0

[1− exp

(−(𝛾3+1)𝛾𝑡ℎ

𝛾1− 𝐶𝛾𝑡ℎ

𝛾1𝛾2

)]×exp

(− 𝛾2

𝛾2

)𝑑𝛾2.

After some manipulations, the integral 𝐼1 (𝛾3) can be ex-pressed as

3The 12

in (7) is due to the fact that the relay operates in the half-duplexmode.

357

𝐼1 (𝛾3) = 1− 1

𝛾2

exp

(− (𝛾3 + 1) 𝛾𝑡ℎ𝛾1

)∫ ∞

0

exp

(−𝐶𝛾𝑡ℎ𝛾1𝛾2

)×exp

(− 𝛾2

𝛾2

)𝑑𝛾2.

(10)Now using [8, eq. (3.471.9)], the above integral can be

written in a closed-form as

𝐼1 (𝛾3) = 1−2

√𝐶𝛾𝑡ℎ𝛾1𝛾2

exp

(− (𝛾3 + 1) 𝛾𝑡ℎ𝛾1

)𝐾1

(2

√𝐶𝛾𝑡ℎ𝛾1𝛾2

)(11)

where 𝐾1 (.) 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

𝐶𝑜𝑢𝑡 = 1− 2𝐿

𝐿+ 𝛾𝑡ℎ

√𝐶𝛾𝑡ℎ𝛾1𝛾2

exp

(−𝛾𝑡ℎ𝛾1

)𝐾1

(2

√𝐶𝛾𝑡ℎ𝛾1𝛾2

)(12)

where 𝐿 ≜ 𝛾1

𝛾3and 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 𝛾2 → ∞),(12) reduces to a simpler expression as follows

𝐶𝑜𝑢𝑡 = 1− 𝐿

𝐿+ 𝛾𝑡ℎexp

(−𝛾𝑡ℎ𝛾1

). (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

𝛾𝑡 =𝛾1𝛾2

𝛾2𝛾3 + 𝐶(14)

where in this case 𝐶 can easily be obtained from [2] as

𝐶 =

[E

(1

𝛾1 + 𝛾3

)]−1

. (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) 𝛾1 = 𝛾3 (𝐿 = 1)and ii) 𝛾1 ∕= 𝛾3 (𝐿 ∕= 1).

A. Case I: 𝛾1 = 𝛾3 = 𝛾 (𝐿 = 1)

Since the ergodic capacity (per unit channel bandwidth),(𝐶𝐸), of the dual-hop network (with channel informationavailable only to the receiving terminals) is given as

𝐶𝐸 =1

2E [log2 (1 + 𝛾𝑡)] . (16)

Therefore substituting (14) into (16) and after re-adjusting wehave

𝐶𝐸 =1

2ln(2)

⎡⎢⎣E (ln (𝑧𝛾2 + 𝐶))︸ ︷︷ ︸

≜𝐴

−E (ln (𝛾2𝛾3 + 𝐶))︸ ︷︷ ︸≜𝐵

⎤⎥⎦ (17)

where ln (.) denotes the natural logarithm and 𝑧 ≜ 𝛾1 + 𝛾3with its probability density function (PDF) given by [10]

𝑓𝑧(𝑧) =𝑧

𝛾2 exp

(− 𝑧

𝛾

). (18)

Now solving 𝐴 in (17), we can have

𝐴 =

∫ ∞

0

[1

𝛾2

∫ ∞

0

ln (𝑧𝛾2 + 𝐶)exp

(−𝛾2𝛾2

)𝑑𝛾2

]𝑓𝑧(𝑧)𝑑𝑧.

(19)The above integral inside the square brackets can be further

solved using [8, eq. (4.337.1)] and the identity −𝐸i (−𝑥) =Γ (0, 𝑥) for 𝑥 > 0

𝐴 =

∫ ∞

0

[ln(𝐶) + exp

(𝐶

𝛾2𝑧

)Γ

(0,

𝐶

𝛾2𝑧

)]𝑓𝑧(𝑧)𝑑𝑧 (20)

where above 𝐸𝑖 (.)and Γ (, ., ) 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

𝐴 = ln(𝐶) +32𝐶

𝛾 𝛾2

S−3,2

(2

√𝐶

𝛾 𝛾2

)(21)

where S𝜇,𝑣 (.) denotes the Lommel function and is availableas a built-in function in Maple.

Similarly (like the derivation of 𝐴), 𝐵 can be written in aclosed-form as

𝐵 = ln(𝐶) + 8

√𝐶

𝛾 𝛾2

S−2,1

(2

√𝐶

𝛾 𝛾2

). (22)

Lastly, substituting (21) and (22) into (17) we get (25).

B. Case II: 𝛾1 ∕= 𝛾3 (𝐿 ∕= 1)

In this case, the PDF of 𝑧 (which is a sum of two non-identical exponential random variables) is given as

𝑓𝑧(𝑧) =1

(𝛾1 − 𝛾3)

[exp

(−𝑧

𝛾1

)− exp

(−𝑧

𝛾3

)]. (23)

Now substituting (23) into (20) and following the same deriva-tion as we did before for (21), we have a closed-form for 𝐴as

358

𝐴= ln(𝐶)+8

(𝛾1 − 𝛾3)

√𝐶

𝛾2

[√𝛾1 S−2,1

(2

√𝐶

𝛾1 𝛾2

)

−√𝛾3 S−2,1

(2

√𝐶

𝛾2 𝛾3

)].

(24)Note that the expression for 𝐵 remains exactly the same as in(22) except 𝛾 is replaced by 𝛾3. 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 (𝐿). As expected, outagecapacity decreases with decreasing the interference level (i.e.,increasing 𝐿). However this trend is more pronounced at lowto moderate values of 𝛾. Moreover, at sufficiently high valuesof 𝛾 and for a fixed value of 𝐿, the outage capacity remainsalmost constant. A similar trend is also observed from Fig. 3in which the outage capacity is plotted for different values of𝛾2. In this figure, it is shown that for fixed values of 𝐿 and𝛾3, the outage capacity remains almost constant at asymptoticvalues of 𝛾2. 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 of𝐿. Again it is evident from Fig. 4 that the ergodic capacityincreases with increasing 𝐿 and for a set of any two values of𝐿, the corresponding gap is more at moderate to high values of𝛾. 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 𝐶 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 gain𝐶.

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

γ (dB)

Outa

ge C

apac

ity (

C out)

L=1.0L=10L=20Simulations

Figure 2. Plot of the outage capacity for different values of 𝛾 using (12)with 𝛾𝑡ℎ = 1, 𝛾1 = 𝛾2 = 𝛾3 = 𝛾 for 𝐿 = 1 and 𝛾2 = 𝛾3 = 𝛾 = 𝑘𝛾1 for

𝐿 = 𝑘 with 𝑘 =

{1

10,1

20

}.

10 20 30 40 50 60 70 80 90 10010-2

10-1

γ2 (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 𝛾2 using (12)and (13) with 𝛾3 = 10 dB, 𝛾1 = 20 dB and 𝐿 = 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 𝐶 which is required in (12). Similar to SectionIV, we consider two scenarios here, i.e., i) 𝛾1 = 𝛾3 (𝐿 = 1)and ii) 𝛾1 ∕= 𝛾3 (𝐿 ∕= 1).

Case I: 𝛾1 = 𝛾3 = 𝛾

Using (6) and [12, eq. (5b)], we can write

𝐶 =

[∫ ∞

0

exp (−𝑠)

(1 + 𝑠𝛾)2 𝑑𝑠

]−1

. (27)

359

𝐶𝐸 =4

ln(2)

√𝐶

𝛾 𝛾2

[4

√𝐶

𝛾 𝛾2

S−3,2

(2

√𝐶

𝛾 𝛾2

)− S−2,1

(2

√𝐶

𝛾 𝛾2

)]. (Case I) (25)

𝐶𝐸 =4

ln(2) (𝛾1 − 𝛾3)

√𝐶

𝛾2

[√𝛾1 S−2,1

(2

√𝐶

𝛾1 𝛾2

)− 𝛾1√

𝛾3

S−2,1

(2

√𝐶

𝛾2 𝛾3

)]. (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

γ (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). 𝛾1 = 𝛾2 = 𝛾3 =𝛾 for 𝐿 = 1 and 𝛾1 = 𝛾2 = 𝛾 = 𝑘𝛾3 for 𝐿 = 𝑘 with 𝑘 = {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

𝐶 =

[1

𝛾exp

(1

2𝛾

)W−1,− 1

2

(1

𝛾

)]−1

(28)

where W (.) is the Whittaker function defined in [8].

Case II: 𝛾1 ∕= 𝛾3

Using (6) and (23), we can write

𝐶 =

[∫ ∞

0

1

𝑧 + 1𝑓𝑧(𝑧)𝑑𝑧

]−1

(29)

where as above 𝑧 ≜ 𝛾1 + 𝛾3.Therefore applying (23), [8, eq. (3.352.4)] and using the

identity−𝐸𝑖 (−𝑥) = 𝐸1 (𝑥) , 𝐶 can be written in a closed-form as

𝐶 =𝛾1 − 𝛾3

exp(

1𝛾1

)𝐸1

(1𝛾1

)− exp

(1𝛾3

)𝐸1

(1𝛾3

) (30)

where 𝐸1 (.) is the exponential integral defined in [8].

APPENDIX II

Here we will derive the closed-form expression of 𝐶 forthe interference-limited scenario and these expressions arerequired in the Section IV to evaluate (25) and (26).

From (15), we have

𝐶=

[1

𝛾1

1

𝛾3

∫ ∞

0

∫ ∞

0

1

(𝛾1 + 𝛾3)exp

(−𝛾1𝛾1

)exp

(−𝛾3𝛾3

)𝑑𝛾1𝑑𝛾3

]−1

.

(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

𝐶 =

[1

𝛾1

2𝐹1

(1, 1; 2; 1− 𝛾3

𝛾1

)]−1

(32)

where 2𝐹1 (., .; .; .) is the Gauss’ hypergeometric functiondefined in [9, eq. (15.1.1)]. Furthermore for a special caseof 𝛾1 = 𝛾3 = 𝛾, we have 2𝐹1 (1, 1; 2; 0) = 1, therefore theabove expression reduces to

𝐶 = 𝛾. (33)

Moreover, for 𝛾1 ∕= 𝛾3 we can use the identity [9, eq. (15.1.3)]and so (32) simplifies to

𝐶 =𝛾1 − 𝛾3

ln (𝐿). (34)

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