[ieee 2014 ieee 15th international workshop on signal processing advances in wireless communications...

The Achievable Average Rate- Outage ProbabilityTrade off Curve In Two-hop Block Fading Channels

Fatemeh Amirnavaei†, Soroush Akhlaghi‡†Dept. of Electrical and Computer Engineering, University of Ontario Institute of Technology, Canada

‡Dept. of Electrical Engineering, Shahed University, IranEmails: [email protected], [email protected]

Abstract—This paper concerns transmission over a two-hop

quasi-static Rayleigh block fading channel in which communica-

tion is made through the use of a Decode and Forward (DF) relay.

It is assumed that the channel state information (CSI) associated

with both hops are only available at the corresponding receivers,

while they are unknown to the transmitters. Considering an

outage event occurs if the destination can not receive a target rate,

the optimal achievable average rate- outage probability tradeoff

curve is obtained through the use of a multi-layer code at both

hops, termed the broadcast strategy. In order to maximize the

average rate, we propose an optimum power allocation scheme of

both hops. Numerical results show that the maximum achievable

average rate for the specific outage probability is obtained when

the threshold layer of both hops are equal.

I. INTRODUCTION

The outage probability is commonly used as a measureof the quality of wireless networks in a fading environment.Although it is mostly desirable to have this probability van-ished, it is not the case happening in practice as worst casechannel conditions act as bottleneck in such cases whichseverely degrades the achievable throughput. Thus, from thenetwork management perspective, it is advisable to keep thisparameter just below a value which can be tolerated by theservice being used. The tradeoff between the achievable rateand outage probability is first introduced for block interferencefading channels in [1]. An information outage occurs when theinstantaneous mutual information is less than a target rate.

In some cases, it is desirable to investigate the scenario inwhich the channel state information (CSIT) is not availableat the transmitter. This may happen if there is no feedbackmechanism or the signal fed back to the transmitter is eithertoo poor or outdated to be incorporated at the transmitter.[2] introduced the so called broadcast approach to maximizethe achievable average rate of single-hop flat fading channel,when there is no CSIT. According to the notion of broadcastapproach, a point to point channel can be translated to a pointto multi-point network with an infinite number of virtuallyordered users in which each virtual user corresponds to achannel strength’s realization. It is widely recognized thatsuperposition code can approach the sum-rate capacity ofsuch a network. Motivated by this, the achievable averagerate of using broadcast strategy in two-hop network withno direct link between transmission ends is studied in [3].Accordingly, various relaying protocols such as Decode andForward (DF), Amplify and Forward (AF), and Quantize and

Forward (QF) are studied there. When there is no direct linkbetween transmission ends, the destination can merely receivethe relay’s transmitted signal which is a degraded version ofthe relay’s. In this case, it is argued that the DF strategyis optimal [4]. Accordingly, using the broadcast strategy, theachievable average rate of two-hop channel in the presence ofDF relay is numerically computed in [4].

To the best of our knowledge, the outage analysis has notbeen studied under multi-layer approach in the relay networks.In this paper, we consider outage probability in DF relay two-hop channel to obtain maximum achievable average rate usingmulti-layer approach.

In the current study, we consider a scenario in which thedestination receives at least a minimum amount of rate toestablish a reliable connection, otherwise an outage eventoccurs. The main goal of this work is to obtain the achievableaverage rate- outage probability tradeoff curve. In other words,for a specific outage probability, it is desirable to computethe optimal power allocation across code layers, in order tomaximize the achievable average rate.

This paper is organized as follows: The system model isintroduced in Section II. Section III presents some backgroundinformation. The proposed method is given in Section IV.Finally, numerical results and conclusion are provided inSections V and VI, respectively.

II. SYSTEM MODEL

We assume the received signal at the relay�y

r

(t)

�is a

faded noisy version of the transmitted signal�x

s

(t)

�, i.e.,

y

r

(t) = h

1

x

s

(t) + n

r

(t). It is assumed n

r

(t) is an AWGNof unit variance, i.e., n

r

⇠ N (0, 1), and h

1

denotes thechannel gain of first hop which is fixed during a transmissionblock. The received signal at the destination is given byy

d

(t) = h

2

x

r

(t) + n

d

(t) which is a faded noisy version ofthe relay’s transmitted signal, i.e.

�x

r

(t)

�. Let h

2

and n

d

(t)

denote, respectively, the channel gain and AWGN of unitvariance of the second hop, i.e., n

d

(t) ⇠ N (0, 1). Similarly,we define channel strength g

i

of the i

th hop as g

i

= |hi

|2 fori = 1, 2. Moreover, the source and relay power are limited byP

s

and P

r

, respectively.We assume that an outage event happens when the channel

strength of each hop is less than a certain threshold. The outageprobability is defined by Prout = Pr(g

1

< s

th1

_ g

2

< s

th2

) =

1�Pr(g

1

� s

th1

^ g

2

� s

th2

), where s

th1

and s

th2

represent

2014 IEEE 15th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

978-1-4799-4903-8/14/$31.00 ©2014 IEEE 329

the thresholds of first and second hop, respectively. In Rayleighflat fading environment, the channel strengths of both hops arestatistically independent and noting Pr(g

i

� s

thi

) = e

�s

thi

for i = 1, 2, the outage probability can be computed as,

Prout = 1� e

�(s

th1+s

th2). (1)

In (1), different values of sth1

and s

th2

may result in the sameoutage probability, as long as the sum s

th1

+ s

th2

remainsconstant. In the following, for a given outage probability, weobtain the optimum values of s

th1

and s

th2

maximizing theachievable average rate. To this end, motivated by the broad-cast strategy approach, in addition to the single layer codewhich aims at sending the target rate R

0

, a multi-layer codingapproach is proposed, to maximize the average throughputfor a given outage probability. This approach guarantees aminimum rate of R

0

at the destination; otherwise an outagehappens.

III. BACKGROUND INFORMATION

According to the broadcast strategy, a point to point channelin the lack of CSIT can be thought as a point to multi-point network in which there are some virtual destinations,corresponding to every possible channel realization. It isargued that multi-layer coding approach can maximize thesum-rate capacity of such a network [2]. In other words,there is a correspondence between code layers and channelstrengths, such that for a given channel realization, all lay-ers indexed by channel strengths below the current channelstrength can be successfully decoded at the destination. Math-ematically speaking, assuming a fractional power ⇢(s)ds isallocated to the layer s, the incremental rate of layer s isdR(s) = log(1 +

s⇢(s)ds

1+sI(s)

) ' s⇢(s)ds

1+sI(s)

, where I(s) is theinterference power arising from layers indexed s+ds to +1,i.e., I(s) =

R1s

⇢(u)du or equivalently ⇢(s) = �I

0(s). Thus,

it follows R(s) =

Rs

0

u⇢(u)du

1+uI(u)

. Also , the average rate is givenby R

bs,ave

=

R10

f(s)R(s) =

R10

(1 � F (s))dR(s), wheref(s) and F (s) denote, respectively, the probability densityfunction (pdf) and the cumulative distribution function (cdf).The original problem can be cast as the following optimizationproblem,

Rbs,ave = maxI(.)

R10

(1� F (s))

�sI

0(s)

1+sI(s)

ds

s. t.R10

⇢(s)ds = I(0) = P,

(2)

and the optimal function I

opt

(s) is analytically derived in [2].

IV. THE PROPOSED METHOD

In this section, the average rate versus outage probabilitytradeoff in a two-hop relay-assisted network is considered. Asit is mentioned before, an outage occurs if g

1

< s

th1

or g2

<

s

th2

. In this case, the achievable average rate is

R

avg

= (1� Pr

out

)R

avg,o

, (3)

where Pr

out

is the outage probability defined in (1). Ravg,o

is the average rate at the destination when g

1

� s

th1

andg

2

� s

th2

. Considering the conditional pdf of the source-relay

and the relay-destination links are denoted, respectively, byf

g1(x|x � s

th1

) and f

g2(y|y � s

th2

), following the sameapproach in [4], the conditional average rate at the destinationcan be formulated as1,

R

avg,o

= E

g1 [Eg2 [Rd

(g

2

= y | Rr

(x))]] (4)

=

Z 1

s

th1

Z 1

s

th2

f

g1(x|x � s

th1

)f

g2(y|y � s

th2

)

⇥Z

y

s

th2

s⇢

r

(s | Rr

(x))ds

1 + sI

r

(s | Rr

(x))

dydx, (5)

where R

r

(x) and ⇢

r

(·|Rr

(x))ds are the received rate and thefractional power allocated to the layer indexed by s at therelay. Also, the conditional achievable rate at the destination isdenoted by R

d

(y | Rr

(x)) =

Ry

s

th2

s⇢

r

(s|Rr

(x))ds

1+sI

r

(s|Rr

(x))

. From nowon, for ease of notation, the conditional pdf of f

g1(x|x � s

th1

)

and f

g2(y|y � s

th2

) are represented by f

g1(x|sth1) andf

g2(y|sth2), respectively. Since the optimization problem isnot in the isoperimetric form, we rewrite it as two-step opti-mization problem and solve by Euler-Lagrange equations [4].

max⇢

s

(·)

Z 1

s

th1

dxf

g1(x|sth1)⇥ (6a)

max⇢

r

(·|Rr

(x))

Z 1

s

th2

dyf

g2(y|sth2)Z

y

s

th2

s⇢

r

(s | Rr

(x))ds

1 + sI

r

(s | Rr

(x))

s. t.Z 1

s

th1

⇢

s

(a)da = P

s

(6b)Z 1

s

th2

⇢

r

(s | Rr

(x))ds = P

r

(6c)

R

T

r

,Z 1

s

th2

s⇢

r

(s | Rr

(x))ds

1 + sI

r

(s | Rr

(x))

R

r

(x), (6d)

where ⇢

s

(a)da is the fractional power of the first hop allocatedto the layer indexed by a. (6b) and (6c) indicate the powerconstraints of source and relay, respectively. The inner max-imization problem is subject to (6c) and (6d). (6d) indicatesthat the total transmitted rate at the relay (RT

r

) can not exceedits received rate, i.e., R

r

(x). In what follows, we first addressthe inner optimization problem given in (6). Then we find theoptimal power allocation of the first hop.

A. Power allocation strategy of second-hop

Assuming R

r

(x) bits received from the first hop, the relayaims at finding the optimal power allocation across codelayers of the second hop to maximize the received rate at thedestination. The optimization problem (6) can be written as,

R

ave

(R

r

(x)) ,max

⇢

r

(·|Rr

(x))

R1s

th2dyf

g2(y|sth2)Ry

s

th2

s⇢

r

(s|Rr

(x))ds

1+sI

r

(s|Rr

(x))

s. t. (6c), (6d).

(7)

This problem can be tackled through the use of Euler-Lagrangeequations. To this end, noting ⇢

r

(s|Rr

(x)) = �I

0r

(s|Rr

(x)),

1Note that [4] does not consider the outage event. In other words, it merelyinvestigates the case of sth1 = sth2 = 0


330

and using the method of integration by parts, the integrand of(7) can be written as,

G(s, I

r

(s), I

0r

(s))= (1� F

g2(s|sth2))�sI

0r

(s | Rr

(x))

1 + sI

r

(s | Rr

(x))

��

sI

0r

(s | Rr

(x))

1 + sI

r

(s | Rr

(x))

. (8)

The Euler equation states that the optimal value of Ir

(s|Rr

(x))

is the solution of the partial differential equation G

I

r

�d

ds

G

I

0r

= 0, where G

I

r

=

@G

@I

r

and G

I

0r

=

@G

@I

0r

. One canreadily verify that the optimal I

r

(s|Rr

(x)) is, Ir

(s | Rr

(x)) =

1�F

g2 (s|sth2)+��sf

g2 (s|sth2)

s

2f

g2 (s|sth2).

Noting in Rayleigh flat fading channel, the conditional pdfand cdf of the second hop can be computed as f

g2(s|sth2) =e

�s

e

�s

th2and F

g2(s|sth2) = e

�s

th2�e

�s

e

�s

th2, respectively. Therefore,

we have

I

r

(s | Rr

(x)) =

e

�s

(1� s) + �e

�s

th2

s

2

e

�s

. (9)

We assume the layers are distributed in the interval[s

0

, s

1

], where s

1

is obtained from the boundary conditionI

r

(s

1

|Rr

(x)) = 0. Also, if the outage probability was set tozero, s

0

would be determined from the boundary conditionI

r

(s

0

|Rr

(x)) = P

r

, meaning the total power allocated tolayers should not exceed P

r

. However, in the presence of anoutage, one should verify whether s

th2

falls in to the interval[s

0

, s

1

] or not. In what follows, we are going to address thisissue.

First, let assume s

0

s

th2

s

1

. Since the layers whichare lower than s

th2

are located in the outage region, the powerassigned to the layers below s

th2

, i.e., Pr

�I

r

(s

th2

|Rr

(x)), isallocated to the threshold layer. Thus, I

r

(s|Rr

(x)) becomes,

I

r

(s|Rr

(x)) =

8<

:

P

r

s < s

th2

e

�s

(1�s)+�e

�s

th2

s

2e

�s

s

th2

< s s

1

0 s > s

1

,

(10)

where s

1

is obtained from the boundary conditionI

r

(s

1

|Rr

(x)) = 0. Note that s

1

depends on the valuesof s

th2

and R

r

(x). Consequently, noting ⇢

r

(s|Rr

(x)) =

�I

0r

(s|Rr

(x)) and I

r

(s

th2

|Rr

(x)) =

1�s

th2+�

s

2th2

, the powerdistribution function becomes

⇢

r

(s | Rr

(x)) = (11)✓P

r

� 1� s

th2

+ �

s

2

th2

◆�(s� s

th2

)� I

0r

(s)1(sth2

< s s

1

).

In order to determine the power allocation strategy (11),we should obtain and substitute the parameters s

1

and �.From the complementary slackness condition, we know thatif the constraint R

T

r

< R

r

(x) in (6d) holds, � is zero,thus I

r

(s|Rr

(x)) =

1�s

s

2 . The solution of I

r

(s

1

|Rr

(x)) = 0

is s

1

= 1 and ⇢

r

(s|Rr

(x)) in (11) is fully characterized.Otherwise, if we have R

T

r

= R

r

(x), according to the slacknesscondition we have � 0, thus one needs to concurrentlycompute both of � and s

1

. To this end, substituting (10) into

R

T

r

=

R1s

th2

s⇢

r

(s|Rr

(x))ds

1+sI

r

(s|Rr

(x))

, it follows,

R

T

r

= �s

1

+ s

th2

+ 2 ln

s

1

s

th2

+ log

✓(1 + s

th2

P

r

)

s

th2

1 + �

◆= R

r

(x) . (12)

Referring to (10), we have,

I

r

(s

1

|Rr

(x)) = 0 ! s

1

= 1�W

L

(�e

1�s

th2�), (13)

where W

L

(·) is the Lambert function.2 Then (12) and (13)can be solved numerically to obtain s

1

and �. Note that theminimum value of � is determined by the marginal case s

1

=

s

th2

(single layer code at the threshold s

th2

). Substituting s

1

=

s

th2

into (12), one can readily get �min

= s

th2

� 1 0.Now, let consider the case in which s

th2

< s

0

< s

1

. Fromthe complementary slackness condition, similar to the firstscenario, when R

T

r

< R

r

(x) in (6d) holds, � should be zero.In this case, referring to (9), the values of s

0

and s

1

are givenby s

0

=

2

1+

p1+4P

r

and s

1

= 1 using the boundary conditions.This case happens as long as we have s

th2

2

1+

p1+4P

r

. Onthe other side, if we have R

T

r

= R

r

(x), � is non positive.Therefore, referring to (9), s

0

and s

1

are obtained from bound-ary conditions I

r

(s

0

|Rr

(x)) =

e

�s0(1�s0)+�e

�s

th2

s

20e

�s0= P

r

and

I

r

(s

1

|Rr

(x)) =

e

�s1(1�s1)+�e

�s

th2

s

21e

�s1= 0, and after some

manipulations, s0

and s

1

need to satisfy,

e

�s0(1� s

0

)� e

�s1(1� s

1

)

s

2

0

e

�s0= P

r

. (14)

Furthermore, using (6d) and noting R

T

r

= R

r

(x), we haveR

T

r

= �s

1

+ s

0

+ 2 ln

s1s0

= R

r

(x). This equation togetherwith (14) result in the optimum values of s

0

and s

1

. Note thatthe values of s

0

and s

1

are independent of sth2

. Finally, whens

th2

> s

1

, all power is assigned only to the threshold layers

th2

by a single layer code.Thus, referring to (7) and after some manipulations, the

average rate at the destination when the available rate at therelay is R

r

(x), becomes

R

ave

(R

r

(x))=

8<

:

R

ave,1

s

th2

< s

0

R

ave,2

s

0

s

th2

s

1

log(1 + s

th2

P

r

), s

th2

> s

1

(15)

where R

ave,1

= e

s

th2(e

�s1�e

�s0)+2e

s

th2(E

1

(s

0

)�E

1

(s

1

)),R

ave,2

= e

s

th2(e

�s1 � e

s

th2) + 2e

s

th2(E

1

(s

th2

) � E

1

(s

1

)) +

log((1 + s

th2

P

r

)

s

th21+�

), and E

1

(s) =

R1s

e

�t

t

dt (s � 0) is theexponential integration function.

B. Power allocation strategy of first hop

Knowing the value of the inner integration, i.e.,R

ave

(R

r

(x)), the equation (6) can be written as

R

avg,o

= max⇢

s

(.)

R1s

th1dxf

g1(x|sth1)Rave

(R

r

(x))

s. t. (6b),(16)

2The Lambert function WL(z) is z = WL(z)eWL

(z) for any complexnumber z.


331

0.05 0.1 0.15 0.2 0.25 0.31.8

1.85

1.9

1.95

2

2.05

2.1

2.15

sth1

Aver

age

achi

evab

le ra

te [N

ats/

Cha

nnel

use

]

Ravg(Prout=0.3)Ravg(Prout=0)

Fig. 1. The achievable average rate versus threshold level of first hop.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Outage probability Prout

Aver

age

achi

evab

le ra

te [N

ats/

Cha

nnel

use

]

RavgRavg(Pr=0)

Fig. 2. The achievable average rate versus outage probability tradeoff curve.

where R

r

(x) =

Rx

s

th1

a⇢

s

(s)da

1+aI

s

(a)

. Considering the in-tegrand of (16) is denoted by S(x, I

s

(x), I

0s

(x)) =

f

g1(x|sth1)Rave

(R

r

(x, I

s

(x), I

0s

(x))), the optimum func-tion I

s

(.) can be tackled through the use of the Euler equation,i.e, S

I

s

� d

dx

S

I

0s

= 0, where S

I

s

=

@S

@R

r

@R

r

@I

s

and S

I

0s

=

@S

@R

r

@R

r

@I

0s

. Thus, in Rayleigh fading environment, noting theconditional pdf of channel strength is f

g1(x|sth1) =

e

�x

e

�s

th1,

we have,

R

0ave

(t)[

Zx

s

th1

da

1� a� a

2

I

s

(a)

(1 + aI

s

(a))

2

] (17)

�R

00ave

(t)[(

�xI

0s

(x)

1 + xI

s

(x)

)

Z 1

s

th1

�ada

1 + aI

s

(a)

] = 0,

where for ease of notation it is assumed t , R

r

(x, I

s

, I

0s

).Also, R

0ave

(·) and R

00ave

(·) denote the first and the sec-ond order derivatives of R

ave

(R

r

(x)), respectively. Notethat I

s

(·) in (17) does not yield an analytical solution,hence, one should rely on the numerical methods. To thisend, I

s

(·) is replaced by N equal-distance discrete points[I

s

(l(1)), I

s

(l(2)), ..., I

s

(l(N))], where I

s

(l(i)) = (P

s

�P

0

)

N�i

N�1

and P

0

is the power assigned to the single-layercode with rate R

0

. Note that R0

is the minimum rate receivedat the destination to guarantee that the outage event will nothappen. Accordingly, N � 1 values of l(i) for i = 2, . . . , N

are recursively determined such that (17) holds for each value.Finally, the power allocation density function ⇢(s) can benumerically derived from ⇢

s

(x) =

�dI

s

(x)

dx

.

TABLE I

sth1 0.0272 0.0972 0.1772 0.2572 0.3372sth2 0.3295 0.2595 0.1795 0.0995 0.0195R

avg,o 2.6214 2.9190 2.9889 2.9187 2.7444R

avg

1.835 2.0433 2.0922 2.0431 1.9211

V. NUMERICAL RESULTS

In this section, we provide the achievable average rateversus outage probability tradeoff curve numerically. The noisepower is set to 0 dBW and the source and relay power areP

s

= P

r

= 20 dBW. The minimum guaranteed rate at thedestination R

0

is set to one nats. Note that different valuesof s

th1

and s

th2

result in the same value of the outageprobability as long as (1) holds. Therefore, for a certain outageprobability, different threshold layers result in the differentaverage rate R

avg

. As an example for Pr

out

= 0.3, Fig. 1illustrates the achievable average rate R

avg

versus s

th1

. Tothis end, different values of threshold layers for the constantoutage probability Pr

out

= 0.3 are computed by (1). For eachs

th2

, the average rate of the second hop is obtained by (15).Then, the conditional average rate of destination is given by(16) for the corresponding threshold layer. Finally, R

avg

iscomputed by (3). Therefore, for a certain outage probability,the achievable average rate of the network is different basedon different values of s

th1

and s

th2

. For Prout

= 0.3, Table Iillustrates different threshold layers, the conditional averagerate R

avg,o

and the average rate R

avg

. As it is shown inFig. 1, there is an optimal threshold level s⇤

th1

(or equivalentlys

⇤th2

) for which the achievable average rate is maximized.For Pr

out

= 0.3, one can verify that s

⇤th1

= s

⇤th2

= 0.178

maximizes the average rate. The results are compared with theachievable average rate when there is no outage probability,R

avg

(Pr

out

= 0), which is proposed in [4]. Similarly,optimal s

⇤th1

and s

⇤th2

for each outage probability can beobtained to maximize the achievable average rate. Fig. 2 isprovided to identify the achievable average rate versus outageprobability tradeoff curve. One can see that the performanceof the proposed method in the low outage probability regionis essentially very close to the optimal achievable average rateof two-hop network in which there is no outage probability[4].

VI. CONCLUSION

In this paper, the achievable average rate versus outageprobability tradeoff curve of DF relaying has been obtained.We assumed that there is no direct link from the transmitter tothe receiver and the channel gains are not available at the cor-responding transmitters, where a multi-layer code was used atboth links. Accordingly, the optimal power allocation strategyacross code layers have been derived in order to maximize theaverage rate. Furthermore, it was shown numerically that fora given outage probability, the maximum achievable averagerate is obtained when the threshold layer of both hops areequal.


332

REFERENCES

[1] S. Shamai L. H. Ozarow and A. D. Wyner, “Information theoreticconsiderations for cellular mobile radio,” IEEE Trans. on Veh. Thech.,vol. 3, pp. 359–378, may 1994.

[2] S. Shamai and A. Steiner, “A broadcast approach for a single user slowlyfading MIMO channle,” IEEE Trans. on Info. Theory, vol. 49, no. 10,pp. 2617–2635, Oct. 2003.

[3] A. Steiner and S. Shamai, “Single-user broadcasting protocols over atwo- hop relay fading channel,” IEEE Trans. on Info. Theory, vol. 52,no. 11, pp. 4821–4838, Nov. 2006.

[4] V. Pourahmadi, A. Bayesteh, and A. K. Khandani, “Multilevel codingstrategy for two-hop single-user networks,” in 24th Biennial Symp. on

Communications, 2008, pp. 115–119.


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