Wireless Netw DOI 10.1007/s11276-012-0440-5
Throughput performance of parallel and repetition coding in incremental decode-and-forward relaying Hirley Alves • Richard Demo Souza • Gustavo Fraidenraich • Marcelo Eduardo Pellenz
Ó Springer Science+Business Media, LLC 2012
Abstract The throughput performance of cooperative repetition and parallel coding in incremental decode-andforward is investigated. Four transmission methods are considered: parallel coding with and without distributed space-time coding (PC-ST and PC, respectively); and repetition coding with and without Chase combining at the destination (RC and SC, respectively). The analysis is based on the mutual information seen at the receiver for each scheme. Exact expressions for the outage probability and throughput for all methods are derived. Both ad-hoc and infra-structured relaying scenarios are investigated. Results show that SC can perform very close to RC, PC and PC-ST in terms of throughput, specially in the case of infra-structured relaying or adequate power and rate allocation. The conclusion is that SC would be a better option
H. Alves R. D. Souza (&) CPGEI, Federal University of Technology (UTFPR), Av. Sete de Setembro, 3165, Curitiba, PR 80230-901, Brazil e-mail:
[email protected] H. Alves e-mail:
[email protected] H. Alves Centre for Wireless Communications (CWC), University of Oulu, P.O. Box 4500, 90014 Oulu, Finland G. Fraidenraich Communications Department, University of Campinas (Unicamp), Campinas, SP 13083-852, Brazil e-mail:
[email protected] M. E. Pellenz PPGIA, Pontifical Catholic University of Parana´ (PUC-PR), Curitiba, PR 80215-901, Brazil e-mail:
[email protected]
in practice, since it requires a simpler receiver than PC-ST, PC, and RC. Keywords Cooperative communications Incremental decode-and-forward Repetition coding Parallel coding Power and rate allocation
1 Introduction It is well known that spatial diversity, achieved by the use of multiple transmit and/or receive antennas, is an effective way to circumvent the effects of fading in wireless communications [1]. However, in some circumstances it may be impractical to deploy multiple antennas due to cost or size limitations. Cooperative communications [2–4], which consider the relay channel introduced by Van der Meulen [5], are an alternative to achieve spatial diversity even with single antenna devices. The relay channel model is composed by three nodes, the source, the relay and the destination. The source broadcasts its information, which is heard by both the relay and the destination. Then, the relay forwards the source signal to the destination, which combines the signals received through the two independent paths (source-destination and relay-destination). Thus, spatial diversity can be obtained even with single antenna devices. Many cooperative protocols were proposed, differing basically on the relay behavior. Probably the two most known cooperative protocols are the amplify-and-forward (AF) and decode-and-forward (DF) [2]. The latter has the variant incremental DF (IDF), which makes use of a return channel between the destination and the other nodes, and outperforms both regular AF and DF [2]. The association of error correcting codes with cooperative protocols,
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known as coded cooperation, showed to considerably decrease the outage probability [6, 7]. Coded cooperation can be roughly divided into two classes: repetition coding, which uses the same encoder for source and relay; and parallel coding, in which source and relay encoders send different parities. It has been shown that parallel coding outperforms repetition coding in terms of outage probability [8–11]. However, in parallel coding the code needs to be specifically designed for the relay channel, while the complexity at the destination increases with respect to repetition coding. Moreover, generalizations of parallel and repetition coding using distributed space time codes [12], in which source and relay retransmit synchronously to the destination, can be defined. In this paper we compare the performance of parallel and repetition coding applied to IDF. The analysis is based on the mutual information seen at the receiver, while the metric utilized for comparison is the throughput (average information rate seen at the destination). The numerical results consider two scenarios: ad-hoc relaying, where the nodes are all at ground level, and infra-structured relaying, where the source is at a fixed position and the relay and destination antennas are at a higher height. Four different transmission strategies are considered: PC (regular parallel coding), PC-ST (parallel coding with distributed spacetime codes), RC (repetition coding with symbol combining—or Chase combining [13]—at the destination), and SC (simple RC without Chase combining1). In RC and SC the effective code rate after the relay retransmission is the same as that of the original source transmission. The PC and PC-ST schemes send additional parity bits in a relay retransmission, so that the effective code rate seen at the decoder decreases, increasing both decoder complexity and the probability of successful decoding with respect to RC and SC. Based on mutual information arguments it can be shown that, at least in terms of outage probability, parallel coding outperforms repetition coding. However, we show that in terms of throughput, the difference in performance among the schemes vanishes in most cases. That is true even if rate and power adaptation are applied independently for each transmission method. In light of the above, and given its simplicity, the use of SC—repetition coding without Chase combining—would be preferable in practice. The SC scheme requires a simpler decoder than RC, PC and PC-ST, both in terms of memory requirements or decoding complexity, while not demanding the hard synchronization 1
In SC, as the receiver does not combine the original and the retransmitted messages, the outage is determined by the link with the best SNR between the source to destination and the relay to destination links. Thus, we can say that the receiver applies ‘‘selection combining’’ between the signals received from the source and from the relay.
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required by cooperative schemes based on distributed space-time codes as PC-ST. That is the main contribution of this paper: to show that the simplest repetition scheme could be used with IDF in a number of scenarios without compromising performance. The rest of this paper is organized as follows. Section 2 presents the system model. In Sect. 3 we include the outage probability and the diversity gain analysis of the four transmission schemes, while Sect. 4 deals with the throughput. Section 5 investigates the impact of power and rate allocation, and Sect. 6 concludes the paper.
2 System model Consider a system with three cooperating terminals: source, relay and destination. The source to destination (SD), relay to destination (RD), and source to relay (SR) channels are all subject to quasi-static Rayleigh fading.2. Different path-loss models are considered, according to the scenario under investigation. We assume perfect channel state information at the receivers. The channel noise is considered to be AWGN (additive white Gaussian noise) with variance N0/2 per dimension. A long-term fading channel [14] is assumed, which means that the channel coefficients are constant during several frames. This model is representative of many modern wireless communication systems with reduced mobility and high transmission rates. Moreover, due to their practical feasibility, we assume half-duplex nodes operating in an orthogonal fashion. After a source transmission, the received signal at the destination is: pffiffiffiffiffiffiffiffiffiffiffiffi ySD ¼ PS cSD hSD x þ wSD ; ð1Þ where PS is the source transmit power, cSD is the path loss in the SD link, hSD is the fading, x is the coded frame to be transmitted representing the information bits u, and wSD is noise. We assume that an error free feedback channel between the destination and the other nodes is available. Therefore, ^ after the source transmission, the destination estimates u from the received codeword ySD, detects possible errors and responds with an ACK/NACK depending on the error verification. If an ACK is received, then the source proceeds with the transmission of the next frame. In the case of a NACK, we assume that the relay participates in the retransmission, unless the relay was not able to decode the source message. Our focus is on the case when only one 2
In this paper all channels are modeled as a random variable from a complex Gaussian distribution with zero mean and unity variance, so that the channel envelope is Rayleigh distributed with unity energy.
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retransmission is allowed. As we show in the end of Sect. 5, allowing for more retransmissions does not improve performance due to the long-term quasi-static fading model. Since in this paper we use the outage probability to analyze the performance of different transmission methods, it is important to say that the outage probability is a theoretical limit, and it is defined for an infinite block length code. However, several works have shown that the outage probability predicts very well the frame error rate of good practical codes, even with relative short block lengths [16–18]. Therefore, we assume that the outage probability is a good approximation of the frame error rate for the methods we analyze next.
3 Outage analysis An outage in the direct transmission occurs when ISD \ R, where ISD is the mutual information in the SD link and R is the attempted information transmission rate. With complex Gaussian inputs and unitary bandwidth, the mutual information is [15]: ISD ¼ log2 1 þ SNRSD jhSD j2 ; ð2Þ where SNRSD ¼ PSNc0SD is the average received signal to noise ratio (SNR) in the SD link. The outage probability is then: n o P SD ¼ P fISD \Rg ¼ P SNRSD jhSD j2 \2R 1 12R
¼ 1 eSNRSD ;
3.1 Repetition coding without chase combining (SC) In SC, after the source transmission the destination verifies if the frame was received correctly or not. Supposing an error, and that the relay was able to decode the source message, then the destination requests for a retransmission and the relay does so. The destination verifies the frame copy received from the relay. If the retransmitted frame is also in error, then the destination declares a failure and the source proceeds with the next data frame. The outage probability of SC after cooperation can be written as: P SC ¼ P fISD \R; IRD \Rg ¼ P SD P RD 12R 12R ¼ 1 eSNRSD 1 eSNRRD : ð4Þ Therefore, when the relay is able to decode the source message, an outage in the SC scheme occurs only if both the SD and the RD link are in outage. The overall outage probability, considering the source and a possible relay transmission, is then: OSC ¼ P SD P SR þ P SD ð1 P SR Þ P RD ¼ P SD P SR þ ð1 P SR Þ P SC : ð5Þ 3.2 Repetition coding with chase combining (RC) In RC the copies of a frame received from the source and from the relay are combined at the destination. In this case, supposing the original source transmission and one relay retransmission, the mutual information seen at the destination after the cooperation can be written as [15, 19]: IRC ¼ log2 1 þ SNRSD jhSD j2 þ SNRRD jhRD j2 ; ð6Þ
ð3Þ
where P f/g is the probability of event /. Similarly, and following the same notation, after the source transmission the outage in the SR link is the same as (3) but replacing SNRSD by SNRSR and |hSD|2 by |hSR|2. The outage in the relay to destination link can be written in a similar manner, just replacing the appropriate variables, and where the relay uses power PR. Unless specified otherwise, we consider that PR = PS. The schemes analyzed in this paper differ in the way the relay behaves in case a retransmission is requested, and in how the destination combines the original source transmission and a relay retransmission. We consider the cases of SC (repetition coding without Chase combining at the destination), RC (repetition coding with Chase combining), PC (parallel coding) and PC-ST (parallel coding with distributed space-time coding). Distributed space-time coding with repetition coding is not considered because, as we show later, RC and SC can be already as good as the most elaborated solutions, as PC and PC-ST. Next, both outage probability and diversity gain are analyzed.
while the outage probability is: P RC ¼ P fnIRC \Rg
o ¼ P SNRSD jhSD j2 þ SNRRD jhRD j2 \2R 1 :
Since SNRRD and SNRSD are both exponential distributed, the sum is gamma distributed [20], therefore P RC can be computed as: 12R 12R SNRSD SNRRD SNRRD e 1 SNRSD e 1 P RC ¼ : ð7Þ SNRSD SNRRD When SNRSD = SNRRD, it can be shown that (7) reduces to: 12R
P RC
eSNRSD ð2R þ SNRSD 1Þ ¼1 : SNRSD
ð8Þ
Thus, in RC the mutual information obtained after the accumulation of SNR has to be greater than the original attempted transmission rate. Moreover, the overall outage probability, considering the source and the relay transmission is:
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ORC ¼ P SD P SR P fISD2 \RjISD \Rg
P PC
þ P SD ð1 P SR Þ P fIRC \RjISD \Rg P SD2 P RC ¼ P SD P SR þ P SD ð1 P SR Þ ; P SD P SD
ð9Þ
P fIRC \R; ISD \Rg P RC ; ¼ P fISD \Rg P SD
since IRC C ISD. Likewise, P fISD2 \RjISD \Rg ¼
ð10Þ
P SD2 P SD
:
3.3 Parallel coding (PC) In PC a retransmission carries new coded bits, so that the receiver can concatenate the originally received frame and the retransmitted frame to form a single frame of lower rate. In that case, after the relay retransmission, what we have in the receiver is an accumulation of mutual information [15, 19]: IPC ¼ log2 1 þ SNRSD jhSD j2 þ log2 1 þ SNRRD jhRD j2 ¼ log2 1 þ SNRSD jhSD j2 1 þ SNRRD jhRD j2 : ð11Þ 2
By defining two new variables Z1 = 1 ? SNRSD |hSD| and Z2 = 1 ? SNRRD |hRD|2, and computing the distribution of z R1 1 the product Z = Z1 Z2 as fZ ðzÞ ¼ jwj fZ1 ;Z2 w; w dw [20], 1
it is possible to write the outage probability as:
P PC ¼ P fIPC \Rg ¼ P Z1 Z2 \2R ¼
1 SNRSD SNRRD
Z2R Z z 1
ðw1Þ
e
SNR
e w
SD
ðwz 1Þ SNR
RD
dwdz:
2
2
ð2R 1Þ ðSNRRD þ SNRSD Þ SNR2RD SNR2SD
ð13Þ
We consider the exact expression (12) in our numerical results. Moreover, the overall outage probability of PC, OPC ; is the same as that in (9), but replacing P RC by P PC : 3.4 Parallel coding with distributed space-time codes (PC-ST) While in the previous schemes the retransmission comes from a single node, in PC-ST both source and relay retransmit in perfect synchronism,3 so that the receiver can concatenate the originally received frame and the combined retransmitted frame to form a single frame of lower rate. After a retransmission from the relay, the mutual information at the receiver is: IPCST ¼ log2 1 þ SNRSD jhSD j2 þ log2 1 þ jSNRSD jhSD j2 þ SNRRD jhRD j2 ¼ log2 1 þ SNRSD jhSD j2 1 þ jSNRSD jhSD j2 þSNRRD jhRD j2 ; ð14Þ where j is the ratio between the power used by the source during the retransmission and the power used by the source during the first transmission. When power and rate allocation are not carried out we consider that both source and relay use power P2S in a retransmission, so that j ¼ 12 : Note that in this case all schemes (SC, RC, PC and PC-ST) are under the same overall power constraint. Following similar rationale as in the case of PC, it is possible to write the outage probability of PC-ST as: PPCST ¼ P ðIPCST \RÞ 1 ¼ SNRRD SNRSD pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jð4zþj2Þþ1j1 Z2R Z2j
z wj1
e
wþ1 SNR
RD
ð15Þ SNRw
ðw þ 1Þ 1
SD
dwdz:
0
1 x
ð12Þ
1
A closed-form high SNR approximation to the PC outage probability can be found using e1/x& 1/x ? 1 in (12). Therefore:
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ð2R ðR lnð2Þ 1Þ þ 1ÞðSNRRD SNRSD þ SNRRD þ SNRSD Þ SNR2RD SNR2SD 1
where ISD_2 is the mutual information and P SD2 is the outage after the destination applied Chase combining into two consecutive source transmissions (since we assume a long term quasi-static fading channel, P SD2 is just 3 dB better than P SD in terms of SNR). Two consecutive source transmissions (of the same frame) may happen in the case that the destination could not decode the frame after the first transmission, and that the relay was also not able to decode the source message, which happens with probability ðP SD P SR Þ: Moreover, P fIRC \RjISD \Rg is the probability that an error occurs after the destination applies Chase combining in the source and relay transmissions, given that an error occurred after the original source transmission. For obtaining (9) we also took into account the fact that: P fIRC \RjISD \Rg ¼
By using e 1 þ 1x we can arrive at a high SNR regime approximation in the same way as for PC. Moreover, the overall outage probability of PC-ST, OPCST ; is the same as (9), but replacing P RC by P PCST : 3
The source and the relay retransmit in perfect synchronism only if the relay was able to decode the original source transmission. Otherwise, the retransmission comes from the source alone.
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For SC, assuming the equal SNR case, and from (4) using exp ð1=xÞ ð1=xÞ þ 1; which is valid for higher values of x: 2 1 2R ; ð17Þ P SC SNR
0
Overall Outage Probability
10
−1
10
PC R=1 PC−ST R=1 RC R=1 SC R=1 Direct R=1 PC R=8 PC−ST R=8 RC R=8 SC R=8 Direct R=8
−2
10
−3
10
which results in a diversity gain of DSC ¼ 2: In a similar manner: 2 1 2R P RC ; ð18Þ SNR
−15
−10
−5
which also results in a diversity gain of DRC ¼ 2: Analogously, for the PC case: 0
5
10
15
20
25
30
35
40
SNRSD (dB)
Fig. 1 Outage probabilities of direct transmission, SC, RC, PC and PC-ST for R = 1 bits/s/Hz and R = 8 bits/s/Hz, in ad-hoc relaying
Figure 1 compares the overall outage probabilities of SC, RC, PC and PC-ST, for the cases where R = 1 bits/s/Hz and R = 8 bits/s/Hz, as a function of the average SNRSD. In Fig. 1 we consider an ad-hoc relaying scenario where the nodes are over a straight line, and the source to destination distance dSD is normalized to unity, while the relay to destination distance and the source to relay distance are related by dRD = 1 - dSR. In particular, in Fig. 1 we assume that the relay is at dSR = 0.5. Moreover, we assume a log-distance path loss model [1], with exponent a = 4. The outage probability of a single direct transmission is also shown in the figure for reference. As we can see, PC outperforms the other methods, while SC is the worst performing scheme. Moreover, the advantage of PC over SC increases with R, being between 2 and 3 dBs. Similar results are obtained for other topologies (for instance, the case of equal SNR). However, as we show in Sect. 4, when a return channel is present (IDF protocol) and the throughput metric is taken into account, the conclusions can differ significantly. 3.5 Diversity gain An important parameter in the analysis of different cooperative transmission schemes is the achievable diversity. The diversity gain [21] defined as log P SNR!1 log SNR
D ¼ lim
ð16Þ
measures the SNR exponent that describes how fast the outage probability can be decreased with the SNR for a fixed data rate. Considering a direct link without any help of the relay, it is easy to see from (3) that D ¼ 1:
P PC
2R ðSNRðR lnð2Þ 1 þ 2R Þ þ R lnð4ÞÞ 4R þ 1 ; SNR3 ð19Þ
we get DPC ¼ 2; while for PC-ST, considering j ¼ 12 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 P PCST 39 3 23þ2R 9 1 þ 23þR 3 48SNR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 42SNR 6 1 þ 23þR SNR þ 22þR ð3 6 log 8 2SNRð3 þ log 64Þ: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii þ 6ð3 þ 2SNRÞ log 1 þ 1 þ 23þR ; ð20Þ which also results in DPCST ¼ 2: Therefore, in terms of achievable diversity, the four transmission methods have the same performance. These results are in agreement with the curves presented in Fig. 1. Note that the SC, RC, PC, and PC-ST methods have the same accentuated decay slope in comparison to the direct link slope. It is important to say that in our scenario, a long term quasi-static fading relay channel, D ¼ 2 is the largest achievable diversity. Moreover, above we consider the equal SNR scenario as it simplifies the derivation, but it does not invalidate the result. The above result on the diversity gain is valid even for a non equal SNR scenario since the diversity gain is an asymptotic measure of the error probability at high SNR. In order to better visualize this statement, if a sum power constraint is imposed, i.e., SNRSD ? SNRRD = SNR, and for the case where SNR ! 1; the diversity gain will be the same. Similar analysis has been done in [22]. Having investigated the outage probability and the achievable diversity, next we investigate the throughput performance of the transmission schemes considered in this paper. 4 Throughput analysis We consider the classical definition of throughput found in [23], which is: ‘‘The throughput of an ARQ system is
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defined as the ratio of the average number of information bits successfully accepted by the receiver per unit time to the total number of bits that could be transmitted per unit time’’. Thus, throughput is the rate of correct information transfer between the source and the destination. For the SC method the throughput can be written as: R P SD ð1 P SR Þ 2 ð1 fISC \RjISD \RgÞ
T SC ¼ Rð1 P SD Þ þ
R P SC P SD ð1 P SR Þ 1 2 P SD R ¼ Rð1 P SD Þ þ P SD ð1 P SR Þ ð1 P RD Þ; 2 ð21Þ
¼ Rð1 P SD Þ þ
SD \Rg is the probability where fISC \RjISD \Rg ¼ P fIPSCf\R;I ISD \Rg that an error occurs after the relay retransmission, given that an error occurred after the source transmission. Note that a retransmission from the source is not considered, since the SD link is in outage and without Chase combining a source retransmission would be useless. By its turn, the throughput for RC is:
R P SD P SR 2 ð1 P fISD2 \RjISD \RgÞ R þ P SD ð1 P SR Þ ð1 P fIRC \RjISD \RgÞ 2 R P SD2 ¼ Rð1 P SD Þ þ P SD P SR 1 2 P SD R P RC þ P SD ð1 P SR Þ 1 : 2 P SD
T RC ¼ Rð1 P SD Þ þ
ð22Þ Note that in the case of RC, differently than in the case of SC, we consider the possibility of a source retransmission when the relay was not able to decode the source message. That is because in RC we assume that the destination applies Chase combining between the original transmitted frame and the retransmitted copy (even if both come from the source), therefore making it possible to decode the combined message even if the SD link is in outage. The throughput of the PC and PC-ST schemes, T PC ; T PCST can be computed in the same way as that of RC given in (22), but replacing P RC by P PC or P PCST ; respectively. Moreover, the throughput of a single direct transmission is only T SD ¼ Rð1 P SD Þ: Next we compare the throughput of direct transmission, SC, RC, PC and PCST methods considering both ad-hoc and infra-structured relaying. The main difference between the ad-hoc and the infrastructured scenarios is that in the ad-hoc scenario the source, relay and destination are regular users, while in
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the infra-structured scenario the relay and the destination are fixed devices deployed by the service provider. As a consequence, in the ad-hoc mode all nodes are at the ground level and it is reasonable to assume a severe path loss model. In the infra-structured mode we assume that the relay and the destination (which is a base station) are at a higher height than the source (which is a regular user, and is usually at ground level). Then, it makes sense to consider a different path loss model taking into account the effect of the antenna heights. As the relay and destination antennas are at a higher height, the link between them should be considerably less severe than the link between the source and relay or between the source and destination. 4.1 Ad-hoc relaying In the case of ad-hoc relaying we consider the same scenario as in Fig. 1, for the best dSR for each scheme, and a log-distance path loss model with exponent a = 4. From Fig. 2, which considers R = 8 bits/s/Hz (left) and R = 1 bits/s/Hz (right), we can see that the advantage of PC and PC-ST over RC and SC in the case of R = 8 bits/s/Hz is of up to 3 dB in the mid to low SNR range. We can also note that such advantage decreases considerably for smaller values of R, as it is also shown in Fig. 2 for R = 1 bits/s/ Hz. Moreover, in the high SNR range the methods have similar performance, while RC and SC perform very close to each other in the whole SNR range. 4.2 Infra-structured relaying In the case of infra-structured relaying we consider the uplink of a sectorized cellular wireless cooperative network, with cell radius q. The destination, which acts as a base station, is located at the center of the cell and the source, which is a mobile station, is at the border of the cell. We assume that the relay is at a fixed location and positioned as described in [24], with the objective of minimizing the mean square distance between the relay and all the possible users in that cell sector. Moreover, since the relay and the destination are deployed by the service provider, their antennas may be at a higher height than the user. The large scale path loss model has to take that into account. In [25] a simple path loss model is introduced, which was obtained from several non line-of-sight measurements carried out in the 2-GHz band in urban areas of Japan, and which is capable of handling heterogeneous path loss conditions. According to [25], the path loss (in dB) between nodes i and j is: cij ¼ 51 8 log10 Hi Hj log10 dij þ 8:4 log10 Hi Hj þ 20 log10 ðfc =2:2Þ þ 14;
ð23Þ
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1
0.9
7
0.8 6
5
Throughput (bits/s/Hz)
Throughput (bits/s/Hz)
0.7
4
3
0.6
0.5
0.4
0.3
2
PC R=8 PC−ST R=8 RC R=8 SC R=8 Direct R=8
1
0
0
10
20
30
PC R=1 PC−ST R=1 RC R=1 SC R=1 Direct R=1
0.2
0.1
0
40
−10
0
SNRSD (dB)
10
20
SNRSD (dB)
Fig. 2 Throughputs of SC, RC, PC and PC-ST schemes for R = 8 bits/s/Hz (left) and R = 1 bits/s/Hz (right), in ad-hoc relaying 8 PC−ST PC RC SC Direct
7
Throughput (bits/s/Hz)
where dij is the distance between the transmitter i and the receiver j, Hi is the antenna height at the transmitter, Hj is the antenna height at the receiver and fc is the carrier frequency in GHz. In particular, we consider that the source antenna height is Hs = 2 m, the relay antenna height is Hr = 30 m, the destination antenna height is Hd = 30 m, the carrier frequency is fc = 2 GHz, and the cell radius is q = 3km. Figure 3 presents the throughputs of SC, RC, PC and PC-ST when R = 8 bits/s/Hz. We can see that PC-ST, PC and RC have the same performance, with a very small advantage over SC in the mid SNR range. We have not found any significantly different behavior by considering other values for R or cell radius in this scenario. The throughputs of SC, RC, PC and PC-ST are so similar in the above results for ad-hoc and infra-structured relaying because, specially in the high SNR, the throughput is dominated by the first term in (21), which is R ð1 P SD Þ; and is the same for all methods. Since we consider incremental relaying, the methods would only differ when the SD link is in outage, something that is likely only in the mid to low SNR region. Note that in the typical outage probability analysis, as that shown in Fig. 1, the overall outage is considered. The overall outage is that defined after the source and the relay transmissions, thus
6 5 4 3 2 1 0
5
10
15
20
25
SNR
SD
30
35
40
(dB)
Fig. 3 Throughputs of SC, RC, PC and PC-ST schemes for R = 8 bits/s/Hz considering infra-structured relaying, with cell radius q = 3 km
hiding the impact of the return channel. That is why the comparison among the different transmission schemes differs if one considers the throughput as metric instead of the overall outage probability. We can conclude that, in the presence of a return channel, the throughput is a more meaningful metric than the outage probability.
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SNR value and for each method. Since the adequate allocation of the available resources improves the performance of any transmission scheme, it is important to investigate if the conclusions of the previous section change or not by doing power and rate allocation. The choice of parameters (power and rate) is such that maximizes the particular throughput. The optimization problem can be formalized as:
8
Throughput (bits/s/Hz)
7 6 5 4 3
max
T
subject to
H ð1 þ jÞPH S þ PR PT Rmin R Rmax
R;PH S
2
PC N=2 PC PC−ST RC SC Direct
1 0
0
5
10
15
20
25
30
35
40
SNRSD (dB)
Fig. 4 Throughput of PC with N = 2 retransmissions for R = 8 bits/ s/Hz and ad-hoc relaying. The throughputs of SC, RC, PC and PC-ST with only one retransmission are also shown
4.3 Multiple retransmissions The analysis carried out so far assumed only one retransmission, either from the source or from the relay. Such limitation is motivated by the long-term quasi-static fading model we consider in this paper. As shown in Sect. 3.5, one retransmission is enough for achieving the maximum diversity available in the channel. In order to illustrate that even further, in Fig. 4 we show the throughput for PC when N = 2 retransmissions are allowed, with R = 8 bits/s/Hz in the ad-hoc setup. The outage and throughput equations for a general number of N retransmissions are determined in Appendix section ‘‘Outage Formulation’’. In the figure we also show the throughput of SC, RC, PC and PC-ST for a single retransmission, as well as the throughput for the direct transmission. From the figure we can see that allowing for an additional retransmission brings very small gains and only in the low SNR region. That is because in the mid to high SNR regions more than one retransmission are rarely needed, and when needed the throughput contribution of an additional retransmission is of only R/ (N ? 1). For smaller R the gains are even less significant. Moreover, in the case of infra-structured relaying there is simply no advantage in allowing additional retransmissions. Next we consider only one retransmission.
5 Power and rate allocation In the previous section we considered that the transmit power and the attempted rate are fixed for all methods. In this section we investigate the impact of adapting source and relay transmit power and the attempted rate, for each
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ð24Þ
where T can be T RC ; T SC ; T PC or T PCST , PT is the total utilized power without power allocation when both source and relay transmit, i.e. PT = 2PS, and j = 0 for RC, SC and PC. The maximization can be performed with respect Note that, in general, the throughput T is to R, PH S , or both. not convex in R; PH S : Moreover, since our goal is to compare the different HARQ schemes, we do not focus on the proposal of a particular power and rate allocation solution, but we resort to numerically efficient algorithms.4 For the rate allocation (RA) we consider that R can vary from Rmin = 1 bits/s/Hz to Rmax = 8 bits/s/Hz. At each SNR value we numerically determine the attempted rate R which maximizes the throughput. For the power allocaH tion (PA), we determine the values of PH S ; and therefore PR H since PH S þ PR ¼ PT ; which maximize the throughput. When PA and RA are carried out at the same time, the two parameters (PH S ; and R) are jointly numerically optimized. In the following, when we refer to SNR, we mean the SNR in case of fixed power and rate. Figure 5, which considers ad-hoc relaying, shows the throughput of RC for the cases where there is power and rate allocation (PA RA), only rate allocation (RA), and only power allocation (PA). The case without resources allocation is also shown for reference. From the figure we can see that by proper allocating power and rate the throughput performance can be greatly improved, specially in the mid to low SNR range, where gains of more than 10 dB are achievable. Note also that PA has a greater impact in the high SNR region, while RA is of more importance in the mid to low SNR region. Very similar results are also found if we assume infra-structured relaying, or different methods (PC-ST, PC and SC). In Fig. 5, when using only power allocation, we consider that the rate utilized by the source and the relay is the maximum allowed rate which in this case is R = 8 bits/s/ Hz. Then, power is allocated between source and relay, but following the constraint that the total utilized power is 2PS. 4 The Matlab function fmincon was used to numerically solve the optimization problem.
Wireless Netw 8 RC RC − PA RC − RA RC − PA and RA
Throughput (bits/s/Hz)
7 6 5 4 3 2 1 0
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SNRSD (dB) Fig. 5 Throughput of RC with rate allocation (RA), power allocation (PA), power and rate allocation (PA RA). The case without resources allocation is also shown. Ad-hoc relaying
When doing rate allocation, we allow the source and relay to use different rates ranging from R = 1 bits/s/Hz to R = 8 bits/s/Hz, while the power used by the source is fixed as being PS and the power used the relay is also fixed as being PS, so that the overall power constraint is the same for the two schemes. Moreover, recall that SNRSD ¼ PSNc0SD , so that PS is directly proportional to the SNR. Then, when using power allocation, what happens in the low SNR region is that the outage probability for that attempted information rate R = 8 bits/s/Hz is too high for any power allocation under the given overall power constraint. As a consequence the throughput is low. When in the high SNR region, we have the opposite. The attempted information rate R = 8 bits/s/Hz is adequate for that SNR, and doing power allocation we can maximize the throughput, by making retransmissions very infrequent (the overall power is allocated almost only to the source, so that the success probability after the first transmission is increased). When using rate allocation, in the low SNR we are able to adapt the attempted information rate to a value which is reasonable to that SNR, so that the outage probability is also low. As a consequence, we can obtain an increased throughput with respect to the case of power allocation only (which utilizes a too high attempted information rate). However, in the high SNR region, even if we allocate the maximum allowed attempted rate, the performance of rate allocation is worse than with power allocation because in the former we cannot allocate the overall power (2PS) to the first transmission, so that retransmissions are more often than in the case of power allocation, reducing the throughput. Figure 6 (left) shows the power PH S that is allocated to the source, for each different scheme, when power and rate
allocation are carried out. The power is normalized with respect to the power that is used by the source when no adaptive power allocation is carried out (PS). Note that PH R ¼ ; so that the greater is the power allocated to the PT PH S source (PH ), the smaller is the power allocated to the relay S (PH ). From the figure we can notice that most of the power is R allocated to the source in all cases. For instance, consider the case of repetition coding (RC) at an SNR of 10 dB, where from the figure PH S ¼ 1:75PS : Therefore, in this case we allocate H H PH ¼ 1:75P S to the source and PR ¼ PT PS ¼ 0:25PS to S the relay. Similarly, Fig. 6 (right) shows the rate that is allocated for both source and relay after the optimization. Considering the same case, repetition coding (RC) at an SNR of 10 dB, both source and relay will use R & 3.8 bits/s/Hz. Next we consider only the best performing case: concurrent power and rate allocation. Figure 7 shows the throughput of PC-ST, PC, RC and SC, when both power and rate allocation are performed, for the case of ad-hoc relaying. The throughputs when there is no resource allocation are also shown. Figure 7 shows that the difference in performance between the schemes decreases considerably when power and rate allocation are carried out. The performances of all methods become very similar. Moreover, in the case of infra-structured relaying, as shown in Fig. 8, the performances of all methods, under power and rate allocation, are practically coincident.5 In addition, when PA and RA are considered, using more than N = 1 retransmissions brings no improvements both in the ad-hoc and infra-structured relaying cases.
6 Final comments We investigated the throughput performance of different cooperative IDF schemes, considering ad-hoc and infrastructured relaying scenarios. Four methods were considered: SC (repetition coding without Chase combining), RC (repetition coding with Chase combining), PC (parallel coding) and PC-ST (parallel coding with distributed spacetime coding). Exact outage probability expressions for all the methods were derived and our results show that SC can perform very close to RC, PC, and PC-ST in terms of 5
One might argue that power allocation between source and relay should not be considered for the infra-structured scenario, since the relay is deployed by the service provider. However, we consider the power allocation between source and relay because in an energy efficiency sense it is interesting to see how well the methods can perform under a limited amount of resources (one could compare the efficiency of ad-hoc relaying to that of infra-structured relaying). Moreover, the general conclusion—that the schemes perform basically the same—simply does not change if one considers only rate allocation in the infra-structured scenario.
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Wireless Netw Fig. 6 Power (left) and rate (right) allocation as a function of SNR for PC, PC-ST, RC and SC. Ad-hoc relaying
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Fig. 7 Throughputs of SC, RC, PC and PC-ST schemes with power and rate allocation. The throughputs without resources allocation are also shown as a reference. Ad-hoc relaying
throughput. The PC scheme performs at least as well as the other methods considered here, but the advantage of PC may be very small or basically non-existent for: (1) infrastructured relaying; (2) low attempted information rates; (3) adequate power and rate allocation. In such cases
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Fig. 8 Throughputs of SC, RC, PC and PC-ST schemes with power and rate allocation. The throughputs without resources allocation are also shown as a reference. Infra-structured relaying
repetition coding (RC or SC) would be a better choice since it has a smaller complexity than parallel coding, while the performance is basically the same. Acknowledgments CAPES (Brazil).
This work was partially supported by CNPq and
Wireless Netw
Appendix: The case of N retransmissions Outage formulation Allowing for additional retransmissions from either the sourcr the relay in the case of SC is not useful since we consider a long-term quasi-static channel and there is no symbol combining at the receiver. Therefore, next we consider only the cases of RC, PC, and PC-ST. The RC outage probability in (7) can be easily generalized to the case of N relay retransmissions as: 12R 12R NSNRRD eNSNRRD 1 SNRSD eSNRSD 1 P RCN ¼ ; SNRSD NSNRRD ð25Þ while for the cases of PC and PC-ST, (12) and (15) can be generalized as, respectively: 1 P PCN ¼ P fIPCN \Rg ¼ NSNRSD SNRRD 1 z ð26Þ Z2R Zz1 SNR1 SNRw SNR1 ð1þw ÞN RD SD e RD dwdz; 1 1 ð1 þ wÞN zN 1 1
0
P SDN P SDNþ1 1 P SDN1 P SDN R P RC P RC2 P SD ð1 P SR Þ þ N þ1 P SD P RC P RCN1 P RCN 1 ; P RCN2 P RCN1
ð28Þ
where P SDN is the outage probability in the SD link after N consecutive transmissions from the source when Chase combining is applied at the destination. Moreover, the above throughput formulation can be rewritten as: R P RC T RCN ¼ Rð1 P SD Þ þ P SD ð1 P SR Þ 1 2 P SD
N X R P SDn P SDnþ1 þ P SR nþ1 n¼1 N X R ½P RCn1 P RCn : ð29Þ þ ð1 P SR Þ nþ1 n¼2 The cases of PC and PC-ST can be easily obtained from above.
and P PCSTN ¼
1 NSNRRD SNRSD 1 1 SNR SNRSD Þw z SNR1 Z2R 1þz Z Nþ1 SNR1RD ðwþ1 ÞN þð SNRRDRD SNRSD RD e ðw þ 1Þ1=N z1N 1
1
0
References
dwdz;
ð27Þ when j = 1. Note that P RCN in (25) for N = 1 is the same as P RC defined in (7). Moreover, we use the notation P RC1 and P RC interchangeably. The same holds for the cases of PC and PC-ST. The generalization of the overall outage probabilities after N retransmissions (therefore including the effect of the SR channel) is quite straightforward and can be easily obtained following the principles used in (9). We skip that here for the sake of brevity. Throughput formulation The throughput for RC when N retransmissions are allowed can be written as: R P SD2 T RCN ¼ Rð1 P SD Þ þ P SD P SR 1 2 P SD R P RC þ P SD ð1 P SR Þ 1 2 P SD R P SD2 P SD P SR þ Nþ1 P SD
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Author Biographies Hirley Alves was born in Lapa, Brazil, in 1986. Hirley Alves received the B.Sc. and M.Sc. degrees from Federal University of Technology, Parana´ (UTFPR), Brazil, in 2010 and 2011, respectively, both in Electrical Engineering. From January 2008 to February 2009 he participated of the Capes/Brafitec undergraduate exchange program at Universite´ de Technologie de Troyes (UTT), France. He is currently both with the Federal University of Technology, Parana´ (UTFPR), Brazil and the Centre for Wireless Communications (CWC) of University of Oulu, Finland. His research interests are in the area of cooperative communications.
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Richard Demo Souza was born in Floriano´polis, Brazil, in 1978. He received the B.Sc. and the D.Sc. degrees in Electrical Engineering from the Federal University of Santa Catarina (UFSC), Floriano´polis, Brazil, in 1999 and 2003, respectively. From March 2003 to November 2003 he was a Visiting Researcher in the Department of Electrical and Computer Engineering at the University of Delaware, USA. Since April 2004 he has been with the Federal University of Technology, Parana´ (UTFPR), Curitiba, Brazil, where he is now an Associate Professor. His research interests are in the area of error control coding and wireless communications. Gustavo Fraidenraich graduated in Electrical Engineereing from the Federal University of Pernambuco, UFPE, Brazil, in 1997. He received his M.Sc. and Ph.D. degrees from the State University of Campinas, UNICAMP, Brazil, in 2002 and 2006, respectively. From 2006 to 2008, he worked as Postdoctoral Fellow at Stanford University (Star Lab Group), USA. Currently, Dr. Fraidenraich is Assistant Professor at UNICAMP, Brazil and his research interests include Multiple Antenna Systems, Cooperative systems, OFDM, and Wireless Communications in general. Marcelo Eduardo Pellenz received the B.Sc. degree in Electrical Engineering from the Federal University of Santa Maria (UFSM), Santa Maria, Brazil, in 1993. He received the M.Sc. and D.Sc. degrees in Electrical Engineering from the Department of Communications (DECOM), State University of Campinas (UNICAMP), Campinas, Brazil in 1996 and 2000, respectively. Dr. Pellenz is currently a Full Professor at the Pontifical Catholic University of Parana´ (PUCPR), Curitiba, Brazil. His research interests include digital transmission, channel and source coding, wireless networks, network performance and traffic modeling.
