DSpace at VNU: Outage probability analysis of cooperative diversity DF relaying under Rayleigh fading

Nguyen, Universit of Teehnology Sydney, Australia Quoe Tuan Nguyen, Vietnam National University Hanoi, Vietnam Trang Cong Chung, Vietnam National University Hanoi, Vietnam Abstract: In t

2011 International Conference on Advanced Technologies for Communications (ATC 2011) Outage Probability Analysis of Cooperative

Diversity DF Relaying under Rayleigh Fading

D.T Nguyen, Universit of Teehnology Sydney, Australia Quoe Tuan Nguyen, Vietnam National University Hanoi, Vietnam Trang Cong Chung, Vietnam National University Hanoi, Vietnam

Abstract: In this paper, we present exact analytical expressions for

the outage probability of cooperative diversity wireless relay

networks operating in various decode-and-forward (DF) protocols

(fixed, adaptive, and incremental relaying) under Rayleigh fading

conditions Current works only analyze the asymptotic behavior of

these protocols, either under high signal-to-noise ratios (SNR) or

under low SNR-Iow rate conditions Our analytical results are

presented in such a way that they can be used for both asymptotic

conditions

Index Terms: Multiple relay channel, achievable rate, decode-and­

forward, partial decoding, linear relaying

1 INTRODUCTION

In the slow-fading environment, once a channel is in deep

fade, message coding is no longer effective in improving

transmission reliability, and cooperative diversity transmission

has proved to dramatically improve the performance of

transmission Upper and lower bounds of the capacity of a general

relay channel were first studied in [1] and this work forms the

theoretical foundation of most reseach work on relay networks

today In this paper, we deal only with the classical three-terminal

relay network using low-complexity cooperative diversity

relaying protocols for ease of potential implementation In these

protocols, relay terminals can process the received signal in

different ways, the destination terminals can use different types of

combining to achieve spatial diversity gain, and source and relay

terminals can use repetition code or other more powerful codes to

cope with low-SNR transmission under heavy fade conditions In

slowly fading channels, the fading is assumed constant over the

length of the message block, i.e the channel is memoryless in the

blockwise-sense, and the strict Shannon capacity of the channel is

well defmed and achievable However, when the system is

constrained by the message decoding delay T and the bandwidth

W is also limited, the requirement 2 WI» 1 cannot be met and

channel parameters cannot be modeled as ergodic or

asymptotically mean stationary random variables and the strict

Shannon capacity is zero [5] In most practical situations, the

channel is non-ergodic and capacity is a random variable, thus no

transmission rate is reliable In this case, the outage probability is

defined as the probability that the instantaneous random capacity

falls below a given threshold, and capacity versus outage

probability is the natural information theoretic performance

measure [5] Consequently, as with many authors, this paper

focuses on delay-limited and non-ergodic scenarios, and evaluates performance of cooperative relaying protocols in terms of outage probability

In this paper, in view of ever lowering cost, flexibility and robustness in noise resistance of digital detection, we concentrate only on decode-and-forward (DF) relaying protocols and ignore the noise propagating amplify-and-forward (AF) relaying Fixed relaying (FR) protocols are those in which the relay is continuously active and are normally used when channel state information (CSI) is not available to the transmitter Selection or adaptive relaying (SR) protocols are designed for better efficiency in low SNR conditions and CSI is available at the relay When the measured SNR falls below a threshold, the relay stops its relaying function and the source simply continues its direct transmission to the destination using repetition coding or other more powerful codes Incremental relaying (IR) protocols are those in which the relay only transmits upon a negative feedback (NACK) from the destination, thus avoiding wastage of bandwidth at high SNRs The information rate of an IR network using DF protocol is a random variable [6] depending on how many times the transmission requires either only one sub-block (i.e when the direct link is not in outage) or two sub-blocks (when the direct link is in outage)

In order to calculate the outage capacity, because of the complexity of the probabilistic analysis involved, most authors [3,

4, 6] resort to the max-flow min-cut theorem [2] to fmd an upper bound for the outage capacity of the relay channel The focus of this paper, however, is the exact formulas for the outage probability of the above three forms of DF relaying protocols As was pointed out in the previous paragraph, the information capacity of relaying networks using incremental DF protocols is a random variable depending on the number of sub-blocks being used for transmission The relationship between outage capacity and outage probability is, therefore, also a statistical relationship [6] A simple comparison of the performance of the three DF protocols based on outage capacity is outside the scope of this paper

In practical wireless sensor networks, power is limited and SNR is usually very low, and the performance of relaying networks in terms of energy efficiency in the low SNR regime becomes essential However, in the low SNR regime, the Shannon capacity is theoretically zero as SNR -+O and is no longer a useful

measure Therefore in [5], a more appropriate metric called

outage capacity is defined as the maximal transmission rate for

which the outage probability does not exceed We expect that

some level of synchronization between the terminals is required

for cooperative diversity to be effective When CSI is unavailable

to the transmitters as in most simple implementations in practice,

coherent transmission cannot be exploited, hence even full-duplex

cooperation, i.e where terminals can transmit and receive

simultaneously, cannot improve the total Shannon capacity of the

network Therefore, in this paper we focus on half duplex

operation

2 SYSTEM MODEL AND DEFINITIONS

Figure 1 shows a simple cooperative diversity relay network

using M relaying branches Because of insufficient electrical

isolation between the transmit and receive circuitry, time-division

half-duplex operation proves to be the safest mode In this paper,

the relays are assumed to operate in the time division mode

having two phases: the relay-receive phase and the relay-transmit

phase; each phase or sub-block is of duration T!2 There is no

correlation between the source transmit signal and the relay

transmit signal, fJ = E[x,xr *] = 0, i.e asynchronous case

y"

Relay h

-h",

y,

Figure 1: Diagram of an M-relay cooperative diversity relaying network

Each message from the source is coded into N symbols;

each symbol occupies a transmission time unit; TI2 is the duration

of time slot reserved for each message, i.e N=T!2 Assume that

the source and the relay each transmits orthogonally on half of the

time slots, under the power constraint liT I�=l Ps[n] � Ps and

liT I�=l Pr i [n] � Prb where Ps and Prj are the transmit power of

the source and of the ith relay, respectively

In the relay-receive phase at time n= I ,2, T12, the source

transmits the complete message (N symbols) to both the

destination and the relays (broadcast mode) in the AF case, but

o n l y to the relays in the DF case, i.e only (la) applies

(lb)

where x, y, n, and P are the normalized transmit signal, i.e

E(lxI2) = I, the corresponding received signal, the additive noise which is modeled as a circularly symmetric complex Gaussian random variable with zero mean and variance ri at the receiver, i.e n <N{O, ri ), and the transmit power, respectively The parameters' double subscript ij is to mean being associated with the channel link from i to j hi) is the channel gains (or loss) from node i to node j, being subject to frequency nonselective Rayleigh fading, and is modeled as independent, circularly symmetric, complex Gaussian random variables with zero mean and variance

f.1i) It is well known that under Rayleigh fading, Ihij 12 and its resulting SNR at the receiver is exponentially distributed

In the decode-and-forward (DF) relaying protocol, the relay detects by fully decoding the entire codeword it receives from the source in relay-receive phase, symbol by symbol, then retransmits the signal, after recoding it, to the destination during the relay­ transmit phase

In the relay-transmit phase at time n=T!2+ I, T!2+2, T, the relays send their signals to the destination and the source may or may not send the signal to the destination depending on the relaying protocol used (multiple access mode) The received signal from the relay is

M

;=1

We defme the instantaneous signal-to-noise ratio (SNR) in the received signal as

Yij = IhijI2 /:: = Ihijl2 YijAWGN

where Yi)AWGN is the SNR of the unfaded A WGN channel

Under Rayleigh fading, SNR in (3) is an independent exponential random variable with expected (average) value

(4)

For convenience, and to be consistent with many papers on the subject, in this paper we simply use SNR to mean YAWGN

In this paper, we present the calculation of exact expressions for the cumulative distribution function (cdj) of instantaneous channel gains of various wireless links in a cooperative diversity relay network and the asymptotic behavior of the cd! of these gains either at high SNRs or at low SNR - low rate conditions The cd! function F,,'j (x) is used to calculate the outage probability, p"�;'t (f lth) of the wireless link between two points i and j having instantaneous channel gain hi) for a given outage information rate threshold, Rth The definition of outage is expressed as

Ph:1I1 (SNR, R'h) = Pr{lhu 12 < JI'h} = Fh, (Jllh) (5)

where the channel gain threshold is defined as

/-I,h = (2(M+I)R" -1 ) 1 SNR and M is the diversity order

There are two asymptotic behaviors associated with fllh +O:

one is for very large SNRs and a given finite outage threshold, R,h,

and the other is for both SNR and Rlh being very small

concurrently In the latter case, R'h is equivalent to the E-outage

capacity CE which is defined as the highest transmission rate for

which outage probability stays smaller than E [5] Therefore the

limits of the cd! as fllh +0 for both asymptotic cases are identical

In power-limited applications such as ad-hoc and sensor

networks, efficient design for low SNR operation is more relevant

At low SNRs, the popular Shannon capacity is theoretically zero

and practically difficult to quantify, and E-outage capacity is more

meaningful

In this paper, we may use either the instantaneous gain of the

fading channel, 1 hu 12 with its average flu, or the corresponding

instantaneous signal-to-noise ratio, rij = hJ SNR , wherever is

convenient

3 OUTAGE PROBABILITY CALCULATIONS

3.1 Outage Probability of Fixed DF Relaying

The maximum average mutual information between the input

and the two outputs, achieved by i.i.d complex Gaussian inputs,

of a repetition-coded fixed DF relaying network is [3]

1 DF = min {�IOg(l + Ysr)' � log(l + Ysd + Yrd) } (6)

The first term represents the maximum rate at which the

relay can reliably decode the source message, and the second term

represents the maximum rate at which the destination can reliably

decode the source message provided the source uses repeated

transmission Requiring both the relay and the destination to the

message reliably results in the smaller of the two rates This limits

the performance of a fixed DF relay to that of the link between

the source and the relay, i.e no diversity gain can be achieved

From (6) the corresponding probability of outage under

exponential fading condition is

PFMHl1th) = Pr(lhFDF 12 � 11th)

= 1 -Pr(lhsr 12 > 11th) Pr({lhsr 12 + Ihrd 12} > 11th)

_11 th [ 1 { ( _ !! l!l.) ( _ !! l!l.)}]

= 1 - e II" 1 - I'sd -I'Td /lsd 1 - e IIsd -/lTd 1 - e IIrd (7)

The result in the last line of (7) can be obtained from (A 1),

(A3) and (A5) of the Appendix

By using the first order approximation e-x;:::;l-x, it can be

shown that

L lmpth +o {p;�� (/-Ith )} = _1 (8)

/-Ith /-Isr

The significance of (8) is that it shows that fixed DF relaying does not achieve diversity gain, i.e at high SNR its outage probability decays as 1/SNR instead of 1/SNR2 This is because it depends entirely on the source-to-relay link to fully decode the source message as has been pointed out in [5]

3.2 Outage Probability of Selection DF Relaying

In the selection DF relaying protocol, when the relay is not able to decode the source message, i.e the source-relay link is in outage, the source simply repeats its transmission on the direct link Thus the maximum average information rate in this case is that of repetition coding The information rate of a selection DF relay network can be expressed as below [4]

ISDF = 1

"210g(1 + Ysd + Yrd)'

YST < Yth

Its outage probability under exponential fading condition is

Ptlf� (11th) = Pr(lhsDF 12 � 11th) Pr(2lhsd 12 < 11th) Pr(lhsr 12 < 11th) + Pr(lhsr 12 � 11th) Pr[i�{lhsr 12 + Ihrd 12} < 11th)

(1 -e-l'thI2I'sd)( 1 -e-I'th/I'sr)

(9)

+ /lsd-/lrd e I'ST f/lSd t (1- e -I'Sd ) - /lrd (1 - e -I'Td) } (10)

The result in the last line of (10) can be obtained from (Al) and

(A3) in the Appendix It can then be shown that the result in [4, equ.22] can be obtained as the second order approximation of the exact result in (10), i.e

/-Ird + /-lSI' 2/-1 sd /-lSI' /-I I'd

3.3 Outage Probability ofIR-DF Relaying

(11)

As pointed out in the Introduction, the information capacity

of relaying using incremental DF protocols is a random variable depending on the number of sub-blocks being used for transmission Its information capacity is difficult to defme and its outage probability, therefore, cannot be simply defined based on a capacity threshold Instead, we calculate outage probability of an IR-DF realying network directly from the defmition of outage condition The system is in outage either when the source­ destination and the source-relay links are both in outage, or when the source-relay link is not in outage, i.e able to decode-and forward, but the accumulation of SNR at the destination of signals

from the source and the relay is not enough to exceed the outage

threshold Thus under exponential fading the outage probability

of an IR-DF relaying wireless network is

P/}IU�-DFCl1th) = PrClhlR_DF12 :::; 11th)

= PrClhsdl2 <l1th)PrClhsrI2 <11th)

+ PrClhsr 12 � 11th) Prl:�{lhsr 12 + Ihrd 12} < 11th)

Jlsd -Jlrd

The result in the last line of (l2) can be obtained from (AI) and

(A3) of the Appendix It can then be shown that the result in [6,

equ 5] can be obtained as the second order approximation of the

exact result in (l2), i.e

Lim {P?Ru�DF (Jlth) } = _1 1_ + 1 = 2Jlrd+Jlsr (13)

Jlth >00 Jl�h Jlsd Jlsr 2JlsdJlrd 2JlsdJlsrJlrd

3.4 Cut-Set Bound on Outage probability

The max-jlow min-cut theorem [2] yields the upper bound of the

capacity, i.e lower bound of outage probability, [1][3] It is a

valid upper bound for a general relay channel with multiple input

and multiple output, we therefore use [1, Theorem 3]

!(X,.x2)

The first term is the information capacity of the broadcast

channel, through the relay from X, to Y2 and Y3 with given X2, i.e

the maximum mutual information between the input Xl and the

two outputs Y2 and Y3, while the second term is the capacity of the

multiple access channel, both directly from the source to the

destination and via the relay, from Xl and X2 to Y3, i.e the mutual

information between the two inputs Xl and X2 and the output Y3•

Thus, the upper bound for capacity, in the case of no

correlation between XI and X2 and equal transmit power from the

source and the relay, is

C+ = min H l og(l + (rsd + rsr))' � l og(l + (rsd + r,d)) } (15)

Equivalently, the cut-set-bound of the end-to-end network gain is

Ihcsl = min {clhsl +Ihsrl\ (Ihsl +Ih,i)} (l6)

The corresponding lower bound of the outage probability

under exponential fading condition is

PC1S(l'th) = 1 - Pr[ Clhsdl2 + Ihs,f)>I'th] Pr[ Clhsdl2 + 1 hrdl 2) >I'th]

{

/lsd /lsr {

1 - /lsd _ /lrd /lsd 1 - e -I1sd - /lrd (1 - e -I1rd)

1 { -i!.J.JJ lIth}

1 - - I1sde IIsd -I1sre ;;:

I1sd -I1ST

1 { lIth lIth} ' - I1sde IIsd -I1rde IITd

The result in (l7) can be obtained by using (A3) and (A5) of the Appendix It can be easily seen that the result in [6, equ 14] can

be obtained as the second-order approximation of the exact result

in (l7), i.e

(18)

Outage Pro b a ilit y of Cooperative Diversity Rela y Net w or k s under Rayleigh Fading

x 10" Mu(sd) = 3 Mu(sr) = 2 Mu(rd) = 1

3 r -�' Io - w-e-r-�s�t - ou - - nd � -"'+" -' ! -+ -� -�

<7- I R-DF Re l a i ng : : : : : -e S l ect i on OF Rela y i ng : : : : :

2.5 - - - �- - - -� - - � - - -: - - � - - -: - - � -

I , , , , , ,

, , , , , ,

, , , ,

2 - � -� - - - - � - -: : -: - -; -

- , , ,

, , , , ,

1 , , I , ,

, , , ,

, , , I , ,

� 1.5 -� -� -� ---: -- -: - -: -

, , , , " ,

, , , , ,

1 -� -� -� - -: :- -� --�-

I "

' "

-� -0.02 0.03 0.04 0.05 0.05 0.07 0.08 0.09 0.1

Mu(th)

Figure 2: Outage probability of two different decode-and-forward relaying protocols and their cut-set lower bound for network realization

4 RESULTS AND CONCLUSIONS

With the ever lowering cost and flexibility of implementation

of digital detection, decode-and-forward (DF) relaying protocols have become more and more popular than their noise propagating amplify-and-forward (AF) counterparts In this paper, we have successfully derived exact expressions for the outage probability

of various versions of DF protocols Figure 2 shows the curves of outage probability as a function of channel gain threshold I'th of two decode-and-forward relaying protocols: Selection DF from (10) and Incremental DF from (l2) The figure also shows the cut-set lower bound of outage probability from (17) of a general multi input-multi output relaying network The outage probability curve for Fixed DF from (7) is not shown in Figure 2 because its value is about two orders larger than those of the other DF protocols shown on the figure We have verified all these results using Monte Carlo simulation A plot of the outage probability for FDF protocol in (7) shows that it is almost a linear function of I'tlt

up to as high as 0.1 This, together with the result in (8), shows

that the FDF protocol does not achieve any spatial diversity gain,

as has already been pointed out in [5] We can also conclude from

Figure 2 that as long as the SDF protocol incorporates a repeat

transmission of message via the direct link if the relay is in outage

(during the relay-transmit phase), its outage probability is lower

(i.e better) than that of the IR-DF counterpart However, this

simple comparison may be unfair to the IR-DF protocol because

its information capacity is a statistical variable and therefore the

defmition of J1lh in (5) for IR-DF does not have the same meaning

as for SDF protocol The IR-DF protocol may perform better on

the basis of power and bandwidth efficiencies

What may seem to be surprising from Figure 2 is that the cut­

set outage probability bound lies slightly above the SDF outage

probability This is because the cut-set bound theory in (16)

applies only to systems with continuous energy flow while in

SDF relaying network, energy flow is discontinuous conditioned

on the outage of the source-to-relay channel

Finally, we should point out that the extension of the work in

this paper to cover M-relay case is not a difficult task, particularly

if we assume that all relays are identical

ACKNOWLEDGEMENT

This work was supported by a research grant from Project

QG.1O.44-TRIGB at the University of Engineering and

Technology, Vietnam National University Hanoi

Appendix 1

Calculation of cumulative distribution function of Combined

i.i.d exponential random variables

A1.1 Single exponential random variable

Let u be an exponential r.v with mean J1u, then

fu (u) = _1_e-u1 ""

II"

Fu (u) = l-e-"I""

By using the approximation e -x "" 1-x , we have

(AI)

un 1' >0

A1.2 Sum of two independent exponential random variables

Let s=u+v, where u, v are two independent exponential r.v's

with mean J1u and J1v, respectively, then from the convolution

theorem

fs(J1)= (fu @ fv )p =_ I _ S: e-x1""e-(P-X)/", dx

J1" J1v e-Jli)J" _e-Jli)Ju J1v -J1u

Hence

0,(J1) = s: fs(x)dx=_I- Vlv (l-e-PIP' )-J1u(l-e-PIP,, ) } (A3)

Jiv - Ji"

By using the approximation e -x "" 1-x + x2 /2., we obtain

_

mll >O 2

2

J.i J.i" J.i v

(A4)

A1.3 Distribution of the Minimum of independent exponential random variables

Let m = min(u, v) where u, v are independent exponential random variables with mean J1u and J1v, respectively Then the cdf ofw is

= I-P(u�jJ.,v�jJ.)=I-P(u�jJ.)P(v�jJ.)

For exponential distributions,

1 1

FM (J.i) = 1- exp {-J.i(-+ -)}

J.iu J.iv

i.e m is an exponential r.v having mean J1m which is

Also from (A2),

-J1m J1u J1v

I { FM(J1) } =_1 �

Ji Jiu Jiv

(AS)

(A6)

(4.13)

Note that the distribution of max( u, v) is not an exponential

random variable

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