This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
SNCC: A Selective Network-Coded Cooperation Scheme in Wireless Networks Cong PENG1, Qian ZHANG2, Ming ZHAO1, Yan YAO1 National Lab for Info. Sci. and Tech., Dept. of EE, Tsinghua University, Beijing, 100084, China 2 Dept. of CSE, Hong Kong University of Science and Technology Email:
[email protected],
[email protected], {zhaoming, yaoy}@tsinghua.edu.cn 1
Abstract— Network-coded cooperation is a new communication paradigm that exploits spatial diversity by pooling distributed “communication” (relay) and also the distributed “computation” resources (coding) of different nodes in a network so as to increase wireless networks system performance. In this paper, we develop a Selective Network-Coded Cooperation (SNCC) scheme suitable for multiple unicast transmission pairs in a single-cell wireless network. The relay node performs network coding on received information from multiple sources before forwarding the network-coded data towards the corresponding destinations, which can decode their desired data from the network-coded data and the data they overhear from the other sources. The performance analysis and evaluation in terms of the single-pair outage probability and diversity-multiplexing tradeoff show that under SNCC any transmission pair achieves a full diversity order of M+1 (M is the number of available relays) while holding a better diversity-multiplexing tradeoff performance than existing cooperative cooperation schemes. Moreover, due to the mechanism that the relay adapts the data on which to perform network coding according to the observed source-to-relay channel quality, SNCC smartly avoids error propagation due to incorporating erroneous data into network-coded data, which is verified by numerical results of the single-pair outage probability. Index Terms—Cooperative communication; network coding; outage probability; diversity-multiplexing tradeoff; error propagation
I. INTRODUCTION Cooperative communication is well known as a powerful technology combating slow fading in wireless medium [1]-[3]. By allowing a set of cooperating relays to forward received information, cooperative communication exploits spatial diversity through cooperation among distributed antennas belonging to multiple terminals in wireless networks. Up to now various protocols for cooperative communication have been proposed [4]-[7]. These schemes either allow the relay to simply amplify and forward or fully decode, recode and forward what it has received from a certain source towards the corresponding destination. Thus a relay can help one single source-destination (s-d) pair at a certain time instance. Recently some people intend to leverage network coding technology (Readers interested in details of network coding can refer to [8][9].) in cooperative communications [10]-[12]. In [10] and [11] the authors propose a cooperation scheme termed adaptive network coded cooperation (ANCC). Their key idea is to match network-on-graph with codes-on-graph to construct efficient linear network codes that can account for the changing and lossy nature of wireless networks. In [12], the authors investigate the diversity gain offered at high SNR by applying network coding to a wireless network containing distributed antenna system (DAS) as well as one that supports user cooperation between users. As far as we know, all existing
work related to combining network coding and cooperative communication are only considered under the simple scenario of multiple transmitters communicating to a common destination. In contrast to the above work, in this paper we propose a cooperation scheme termed Selective Network-Coded Cooperation (SNCC) which is suitable for the scenario of multiple sources transmitting to multiple destinations. In SNCC, the relay first selects received information from multiple sources which have source-to-relay wireless channels with good quality, and then performs network coding on these information before forwarding (broadcasting) the network-coded data towards the corresponding destinations. The destinations can recover their desired data from the coded data and the original data they have overheard from other sources (see Section II and III for details). Clearly, in this way a relay can help multiple s-d pairs concurrently. Next, we analyze the outage behavior of SNCC and derive the close form of single-pair outage probability as a function of signal-to-noise ratio (SNR). It is shown that under SNCC each s-d pair obtains a full diversity gain of M+1 (M is the number of available relays in the network) at high SNR. Numerical results of single-pair outage probability further verify that due to the selection mechanism of SNCC, the outage of any source-to-relay channel will not influence the correctness of the eventually formed network-coded data, thus avoiding unnecessary error propagation. We then prove that SNCC outperforms existing cooperation schemes without network coding technology incorporated [4]-[7] in terms of diversity-multiplexing tradeoff. Specifically speaking, SNCC entails less performance loss in spectral efficiency to achieve the same diversity gain than existing cooperative communication schemes such as space-time coded cooperation, opportunistic relaying, etc. The rest of this paper is organized as follows. In Section II, a motivation example is presented. System model as well as the description of SNCC is presented in Section III. In Section IV the outage behavior of SNCC is analyzed and the numerical results are shown. The diversity-multiplexing tradeoff performance of SNCC is investigated in Section V. Section VI concludes the paper and points out some future work. II. MOTIVATION EXAMPLE In this section we show the benefit of SNCC by an example network composed of two s-d pairs and a relay r. In Fig.1, s1 needs to send D1 to d1, and s2 needs to send D2 to d2. When s1 (s2) transmits data D1 (D2) to d1 (d2), d2 (d1) hears D1 (D2).
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
model α s d , α s r and α r d , the channel gains from si to di, si to rm, and rm to di as zero-mean, independent, circularly symmetric complex Gaussian random variables with variances 1 / β s d , i i
i m
m i
i i
Fig.1. An example network showing the benefit of SNCC.
To assure transmission orthogonality, we allocate non-overlapping time slots for different transmissions as in [4]. Fig.2 illustrates the time-division channel allocation of SNCC for this example network.
Fig.2. Example time-division channel allocation for SNCC. We focus on orthogonal transmissions throughout the paper.
As shown in Fig.2, three time slots are needed to complete the whole cooperation process of the two s-d pairs. During the first two time slots, each source transmits its data directly to its destination one by one. If both s1-to-r and s2-to-r channels are good so that r receives both D1 and D2 successfully, during the third time slot r will forward D1⊕D2 (Here we consider xor operation as a simple network coding scheme.) towards the two destinations simultaneously. Since d2 and d1 have already obtained a copy of D1 and D2 during the first and second time slots, respectively, d1 can decode its wanted data D1 from D2 and D1⊕D2, and d2 can decode its wanted data D2 from D1 and D1⊕D2. If only one of the s1-to-r and s2-to-r channels is good so that only D1 or D2 is successfully decoded by r, r will simply forward D1 towards d1 or forward D21 towards d2. Intuitively it can be seen that here a diversity of 2 is provided for each s-d pair. Specifically speaking, for s-d pair i (i=1, 2), the two paths, si-to-di and si-to-r-to-di paths, must be both bad so that Di can not be correctly obtained at di. The mechanism that r dynamically adapts forming the network-coded data based on the observed instantaneous source-to-relay channel quality can avoid unnecessary error propagation. In other word, the outage of any source-to-relay channel will not induce erroneous network-coded data. For example, whether s-d pair 1 successfully completes data transmission eventually will not be influenced by the bad quality of the s2-to-r wireless channel. III. SYSTEM MODEL AND DESCRIPTION OF SNCC
1 / β sirm and 1 / β rm di respectively. Then | α s i d i |2 , | α s i rm |2 and | α rm d i |2
are exponentially distributed with parameter β s d , β s r and βr d , respectively. These channel gains capture the effects of i i
i m
m i
path loss, shadowing, and frequency non-selective fading. The additive channel noise Z at the receiver, is modeled as a complex Gaussian random variable with variance N0. The transmit power for each terminal is assumed to be common and denoted as P. The transmit SNR is denoted as ρ, then we have ρ=P/N0. Each s-d pair attempts a common rate (i.e., spectral efficiency) of R bits per channel use (bit/s/Hz). B. Description of SNCC In this section we describe SNCC in details. For simplicity, xor operation is used as the network coding scheme. A relay selection criterion among multiple relays, which is similar with that in [6] is adopted. Specifically, a single “best” relay is selected out based on the end-to-end instantaneous wireless channel gains from the M relay candidates to act as the network-coding relay for all the N s-d pairs. Set (1) hm = min{| α s r |2 , , | α s r | 2 , | α r d |2 , , | α r d |2 }, m = 1, … , M . 1 m
N m
m 1
m N
The relay node among the M relay candidates that maximizes hm is defined as the “best” relay r, namely, (2) r = arg maxr {hm}. m
A time-division channel allocation of SNCC suitable for the network is presented in Fig.3.
Fig.3. A channel allocation across time for a network composed of N s-d pairs.
As shown in Fig.3, the operation of SNCC includes two phases, Direct Transmission Phase (DTP) and Selective Network-Coding-and-Forwarding Phase (SNCFP). During DTP, each si transmits Di directly to di one by one. During SNCFP, r xors data received during DTP from those sources having good source-to-relay wireless channels and then forwards (broadcasts) the obtained xored data simultaneously to the corresponding destinations. To judge whether a si-to-r channel is good or not, r makes decision based on observed α s r , i
A. System Model Consider a single-cell size network composed of N source-destination pairs denoted as s1-d1, …, sN-dN, and M relays denoted as r1, …, rM. We assume when sj transmits data Dj to dj, di (i≠j) can hear Dj from sj reliably. Suppose wireless channels from si to di, si to rm, and rm to di (i=1, …, N, m=1, …, M) suffer frequency nonselective fading and additive noise. We 1
Actually Di can be seen as the network-coded data obtained by only coding itself.
the instantaneous si -to-r channel gain. If the measured | α s r |2 falls below a certain threshold (this means that the si-to-r channel suffers an outage), r will not incorporate the data from si into network-coded data, and otherwise it will. Since r always selects those data transmitted through good source-to-relay channels on which to perform network coding and to form the forwarding data, we term the scheme Selective Network-Coded Cooperation (SNCC). Under the assumption that when sj transmits data Dj to dj, di (i≠j) can hear Dj from sj reliably, for pair i with good si -to-r channel, di can recover its i
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
desired data from the xored data and the original data it overhears from other sources. In this manner, a complete round of cooperation for N s-d pairs requires only N+1 time slots, the first N time slots for DTP and the (N+1)th time slot for the SNCFP. IV. OUTAGE PERFORMANCE ANALYSIS AND NUMERICAL RESULTS A. Outage Behavior Analysis In this section, we analyze the outage behavior of SNCC. For a certain s-d pair i, we consider it suffers an outage when an instantaneous information rate of R can not be supported. Denote pi as the outage probability of s-d pair i, then we have the following theorem. Theorem 1 For a single-cell size wireless network composed of N s-d pairs and M relays under SNCC, pi, the outage probability of s-d pair i, is given as τ τ − x pi = ∫ 1 − exp − β si di f|α rdi |2 ( x )dx 1 − F|α sir |2 (τ ) 0 + x 1 ρ
(
)
(3)
I i ( X;Y )
) )
( (
(
2 N N N + 1 log 1 + ρ α si di + N + 1 log 1 + ρ α rdi = 2 N log 1 + ρ α , si di N + 1
εq
2
), i ∈ I
q
i ∉ Iq.
(9) Note that the mutual information is divided by (N+1)/N because N+1 time slots are assigned to N s-d pairs. Substituting (9) into (8), there is
(
pi = Pr | α si di |2 + | α rdi |2 + ρ | α sidi |2 | α rdi |2 ≤ τ
( = Pr (| α
+ Pr | α si di |2 ≤ τ si d i
)∑
q ,i∉I q
)∑
q ,i∈Iq
Pr ( ε q )
Pr ( ε q )
) (
|2 + | α rdi |2 + ρ | α si di |2 | α rdi |2 ≤ τ Pr | α si r |2 > τ
(
) (
+ Pr | α si di | ≤ τ Pr | α si r | ≤ τ 2
2
)
)
τ τ − x = ∫ 1 − exp − β si di f|αrdi |2 ( x )dx 1 − F|αsir |2 (τ ) 0 ρ x 1 +
(
(
(
+ 1 − exp − β si di τ
(10)
)
)) F (τ ) . |α sir |2
where F (⋅) and f (⋅) , the CDF and PDF of | α s r |2 and |α | |α |
B. Numerical Results of Outage Behavior In Fig.4 we illustrate the single-pair outage probability which is calculated by (3) for a wireless network composed of N(=3) s-d pairs and M (=1, 2, 3) relays. Here it is assumed that | α s d |2 ,
| α rd | , are decided by
| α s i rm |2 and | α rm d i |2 are all exponentially distributed with a
(
si r
)) F (τ ) , |α sir |2
2
rdi
2
i
2
i
F|α
τ
si r |
τ
+∫
0
2
(τ ) = ∫
0
∑
M m =1
φ
∑ ∫ {( β M
m =1 0
hm
{β sirm e
− β hm φ
− β si rm )e
∏
j ≠m
− ( β hm − β sirm )θ
and f |α
(τ ) = ∑ m =1{β rm di e M
2 rdi |
− β hmτ
τ
+ ∑ m =1 ∫ {( β hm − β rm di )e M
∏
0
∏
(1 − e
j≠ m
− ( β hm − β rmdi )θ
respectively, note that
(1 − e
− βhjφ
j ≠m
− β h jτ
)}dφ
(1 − e
(4)
− βh j θ
)}dθ ⋅ β si rm e
− β si rm φ
)}
∏ j ≠ m (1 − e
dφ
(5) −βhjθ
)}dθ ⋅ β rm d i e
− β rmd iτ
10 10
(6)
βh = ∑n=1 (β s r + β r d ), m = 1,…, M .
(7)
and
N
n m
m n
Proof: First of all, by Lemma 1 ((25)~(26)) in the Appendix, we can obtain F (⋅) and f (⋅) , the CDF and PDF of | α s r |2 |α si r | 2
|α rdi |2
i
and | α rd |2 , as shown in (4) and (5), respectively. For the set {1, …, N} which has Q=2N subsets (For example, if N=2, then Q=4, and the Q subsets are NULL set, {1}, {2}, {1, 2}.), we denote its Q subsets as I1, …, IQ. Let εq denote the event that those wireless channels from si (i∈Iq) to r have their channel gains good enough to support data transmission with no error while those channels from si (i∉Iq) to r do not. Then apparently pi can be formulated as i
pi = ∑ q =1 Pr ( Di is not recovered at d i ε q ) Pr ( ε q ) Q
(
= ∑ q =1 Pr I i ( X;Y ) Q
εq
common parameter 1. Due to symmetric property, it is easy to know that each s-d pair must have a common outage probability. Consequently, we only plot the outage probability of an s-d pair among the three pairs in Fig.4. It illustrates that SNCC enables a full diversity order of M+1 for each s-d pair.
,
τ =(2(N+1)R/N−1)/ρ m
i i
)
(8)
≤ R Pr ( ε q ),
where I i ( X;Y ) , the mutual information between si and di ε q
when event εq happens, is calculated as
Single-Pair Outage Probability
(
+ 1 − exp − β si di τ
10 10 10 10 10 10 10
0
-1
-2
-3
-4
-5
-6
Direct SNCC:N=3,M=1 SNCC:N=3,M=2 SNCC:N=3,M=3
-7
-8
0
5
10
15
20
25
SNR (dB)
Fig.4. Single-Pair Outage Probability versus SNR for a network composed of 2 , N(=3) s-d pairs and M (=1, 2, 3) relays. | α | α s i rm |2 and | α rm d i |2 si d i | (i=1, …, N, m=1, …, M) are all exponentially distributed with parameter 1, R=1 bit/s/Hz.
For SNCC, the relay does not incorporate the data it receives from si into the network-coded forwarding data if the si -to-r channel condition is not good enough to guarantee accurate data transmission. In this way, it can be foreseen that unnecessary error propagation due to coding erroneous data into network-coded data can be avoided. To verify this, in Fig.5 we illustrates the outage probability of each s-d pair calculated by (3) for a wireless network composed of two s-d pairs and one
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
relay r, where the s1-to-r and s2-to-r channel gains have different statistic properties. For comparison, we also plot the outage probability for each pair under network-coded cooperation where the “best” relay r does not invoke the selection mechanism. For this scheme which we call NCC [17], since the relay r always performs network coding on what it receives from both sources, the outage of any si -to-r channel can result in error of the network-coded data. Consequently, replacing F (τ ) in (3) by 1 − ∏ N {1 − F (τ )} [17], we can |α si r |2
|α sir |2
i =1
calculate the single-pair outage probability under NCC. 10
Single-Pair Outage Probability
10
10
10
10
0
-1
-2
1 τ − | α si di |2 + Pr | α si di |2 ≤ τ Pr | α si r |2 ≤ τ pi ≤ Pr | α rd i |2 ≤ 2 1 ρ | α | + s d i i 2 τ − | α si di |2 ≤ Pr hr ≤ + Pr | α si d i |2 ≤ τ Pr ( hr ≤ τ ) 2 1 ρ | α | + s d i i 3 τ τ − x = ∫ 1 − exp −β si di f h ( x)dx + 1 − exp − β si di τ Fhr (τ ) , 0 1+ ρ x r
(
-3
s 1-d1 and s 2-d2, Direct Communication s 1-d1, NCC(w/o Selection Mechanism) s 2-d2, NCC(w/o Selection Mechanism)
-4
5
10
15
20
25
SNR (dB)
Fig.5. Single-Pair Outage Probability versus SNR for a network composed of two s-d pairs and one relay. | α |2 and | α |2 (i=1, 2) are both si d i
rd i
exponentially distributed with parameter 1. | α | s1r
2
and | α | s2 r
2
are
exponentially distributed with 1 and 4, respectively. β < β , which means s1r s2 r that the s1 -to-r channel condition is better than the s2-to-r channel condition in the sense of statistics. R=1 bit/s/Hz.
)
(14)
)
(
-5
0
) (
(
s 1-d1, SNCC s 2-d2, SNCC
10
Diversity-multiplexing tradeoff is proposed by Zheng and Tse [15] for point-to-point MIMO channels, and has been widely used as a powerful performance metric for wireless communication systems, including non-cooperative [16] and cooperative systems [6][13][14]. We recall its definition here. Definition 1 A scheme is said to achieve spatial multiplexing gain Rnorm and diversity gain D if the data rate R satisfies limρ→∞R(ρ)/logρ=Rnorm (11) and the average error probability pe satisfies limρ→∞log pe(ρ)/logρ=−D. (12) We can see that the diversity-multiplexing tradeoff is essentially a tradeoff between the error probability and the transmission rate, just as mentioned in [15]. Theorem 2 In a single-cell size wireless network composed of N s-d pairs and M relays, SNCC achieves a diversity-multiplexing tradeoff characterized by D(Rnorm)=(M+1)(1−(N+1)Rnorm/N), Rnorm∈(0, N/(N+1)). (13) Proof: For pi, the outage probability of any s-d pair i, based on (10) there is
(
))
where hr = min{| α s1r |2 ,
,| α sN r |2 ,| α rd1 |2 ,
(15)
,| α rd N |2 }
with r being the “best” relay decided by the criterion (2), and the function fhr(⋅) and Fhr(⋅) denote the PDF and CDF of the r.v. hr, respectively. Note that in step 2 of (14) we employ the fact that hr ≤| α rd |2 due to (15) and use Lemma 2 in the Appendix. i
Apparently, both SNCC and NCC achieve a full diversity of 2 on both pairs. However, SNCC enables each pair to obtain a reduced outage probability compared with NCC. Moreover, although the relay may perform network coding on the data received from pair 1 and 2, the two pairs do not influence mutually. For example, pair 1 with a better source-to-relay channel condition maintains a lower outage probability than pair 2 with worse source-to-relay channel condition. This is due to the selective network-coding-and-forwarding mechanism at the relay utilized by SNCC, by which unnecessary error propagation is smartly avoided. In contrast, under NCC, because the relay always performs network coding on received information from both sources, the outage probability of each pair is influenced by not only its own source-to-relay channel quality but also the other pair’s source-to-relay channel quality. This results in the same single-pair outage behavior for both pairs, which is higher than that of SNCC. V. DIVERSITY-MULTIPLEXING TRADEOFF In this section, we investigate the diversity-multiplexing tradeoff of SNCC. The result shows that SNCC provides a better diversity-multiplexing tradeoff performance than existing cooperation schemes without employing network coding technology [4]-[7]. This verifies that SNCC is a spectrum efficient cooperation scheme.
Furthermore, due to Lemma 1 ((27)) and Lemma 3 in the Appendix, for fhr(⋅) and Fhr(⋅) we have M (16) f ( x ) = ∑ β exp(−β x)∏ (1 − exp(−β x )) hr
m =1
hm
hm
hj
j≠ m
and
(17)
Fhr ( x) = ∏m =1 (1 − exp(− β hm x)), M
(N+1)R/N
where β h is decided by equation (7). Set t=ρτ=2 based on (16) and (7) there is m
lim
∫
τ
0
−1,
τ − x 1 − exp −β si di f ( x )dx 1 + ρ x hr
τ M +1 τ (1 − ω ) 1 − exp( − β s d ) 1 1 + tω lim f h (τω ) d ω = ∫ lim 0 τ →0 τ → 0 τ M −1 τ 1 M 1− ω {∑ m =1 β h ∏ j ≠ m β h ω}d ω = ∫ βs d 0 1 + tω M −1 1 (1 − ω )ω M N dω = M β s d ∏ j =1 (∑ n =1 ( β s r + βr d )) ∫ 0 1 + tω τ →0
i i
i i
r
m
i i
n j
(18)
j
j n
≤ β si di ∏ j =1 ∑ n=1 ( β sn r j + β r j dn ) , M
N
note that in the first step of (18) we employ the change of variables ω=x/τ, and in the last step of (18) we exploit the bound M −1 1 (1 − ω )ω 1 1 (19) dω ≤ . ( M − 1)! ∫0 1 + t NCC ω
M!
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
On the other hand, based on (17) and (7), there is lim
(1 − exp ( −β τ )) F si d i
hr
VI. CONCLUSION AND FUTURE WORK
(τ )
τ M +1 M 1 − exp( −β s d τ ) ∏ m=1 (1 − exp(− β h x)) = lim lim τ →0 τ →0 τ τM τ →0
m
i i
(20)
= β sidi ∏ m =1 ∑ n =1 ( β sn rm + β rm dn ). M
N
Due to (6), we know τ→0 as ρ→∞. Consequently, (18) and (20) respectively yield the high SNR (ρ) approximation of
∫
τ
0
τ − x 1 − exp − β si di f h ( x)dx 1 + ρx r
•
(21)
=((2( N +1) R / N − 1) / ρ ) M +1 β sidi ∏ m =1 ∑ n =1 ( βsn rm + β rm dn ) M
N
and
(1 − exp ( −β τ ) ) F si di
hr
•
(τ )
(22)
=((2( N +1) R / N − 1) / ρ ) M +1 β si di ∏ m =1 ∑ n =1 ( β sn rm + β rm dn ) . M
N
Suppose (23) R=Rnormlogρ, substituting (21)~(23) into (14), easily we obtain the high SNR (ρ) asymptotic bound of pi as •
pi ≤ ρ
(
( N +1) Rnorm −1)( M +1) N
⋅ 2β si di ∏ m =1 ∑ n=1 ( β sn rm + βrm dn ) . M
N
(24)
Namely, we obtain the diversity-multiplexing tradeoff of SNCC as shown in (13). ■
Fig.6. Diversity-Multiplexing Tradeoff Comparison. SNCC achieves a better diversity-multiplexing tradeoff performance than existing cooperative communication schemes such as space-time coded cooperation [5], opportunistic relaying [6], etc. SNCC can be effectively utilized for a much broader range of Rnorm than the other communication strategies.
We plot the diversity-multiplexing tradeoff of SNCC in Fig.6, in which the diversity-multiplexing tradeoff performance of existing cooperative communication strategies without employing network coding technology is also given. Clearly, SNCC has a better performance in terms of diversity-multiplexing tradeoff than communication strategies such as space-time coded cooperation [5] and opportunistic relaying [6], the diversity-multiplexing tradeoff of which is characterized by D(Rnorm)=(M+1)(1−2Rnorm). This advantage is due to the employment of network coding technology, which makes it possible for a relay to help multiple s-d pairs simultaneously, thus resulting in requiring less bandwidth resources to complete a complete round of cooperation for multiple s-d pairs under SNCC.
It is well known that a key challenge in the area of cooperative communication is to develop protocols powerful in combating slow channel fading at low spectral efficiency cost. In this paper, we propose a cooperation scheme exploiting network coding technology, which is termed Selective Network-Coded Cooperation (SNCC) and suitable for the scenario of multiple sources transmitting to multiple destinations in a single-cell wireless network. SNCC has two main characteristics. First, it endows network coding capability with the relay, i.e., allows the relay to perform network coding on the data received from multiple sources to form the forwarding network-coded data towards the corresponding destinations. Second, the relay only selects the data from those sources having good source-to-relay wireless channels to form network-coded data so as to further improve the system performance. The theoretical analysis and numerical results of the outage behavior show that under SNCC any pair achieves a full diversity order of M+1, with M being the number of available relay candidates. Moreover, since the relay adapts whether or not to incorporate received data into the forwarding data based on the observed source-to-relay channel quality, unnecessary error propagation can be avoided in SNCC, which is verified by numerical results. Finally, we prove that SNCC holds a better diversity-multiplexing tradeoff than existing cooperation schemes that do not utilize network coding technology. This indicates SNCC is powerful in combating fading while entailing less performance loss in spectral efficiency. Note that all the analysis in this paper are based on the assumption that each di can reliably hear Dj (j≠i) when Dj is transmitted from sj to dj. Consequently, in order to fully utilize the benefits of SNCC, one interesting future work is to carefully investigate how to dynamically group multiple s-d pairs together. A straightforward way can be grouping those pairs for which the channel condition between sj (j≠i) and di is constantly good during the communication when performing SNCC. APPENDIX Lemma 1 Suppose for some random variables, Ai, Ai1, …, AiK (i=1,…,M), Ai1, …, AiK are independently exponential distributed with parameter βi1, …, βiK respectively, Ai=min{Ai1, …, AiK}. Let r=argmaxi (Ai ), then for FA (x) and f A (x) , the rk
rk
CDF and PDF of Ark, k=1, …, K, there are FArk ( x) = ∫
x
0
∑
φ
∫ ∑ ∫ {(β x
0
M
i =1 0
i
M i =1
{β ik e − βiφ ∏ j ≠i (1 − e
− β ik )e
− ( β i − β ik )θ
− β jφ
)}dφ +
(∏ j ≠i (1 − e
− β jθ
(25)
))}dθ ⋅ β ik e
− β ik φ
dφ
and f Ark ( x) = ∑i =1 ( βik e − β i x ⋅ ∏ j ≠i (1 − e M
x
+ ∑i =1 ∫ {( βi − βik )e M
0
− ( β i − β ik )θ
−β jx
))
(∏ j ≠i (1 − e
(26) − β jθ
))}dθ ⋅ βik e
− β ik x
,
and for FAr(x), the CDF of Ar, there is FAr(x)=Πi=1M(1−exp(−βix)), where βi=∑k=1Kβik, i=1, …, M. Proof: Since Ai=min{Ai1, …, AiK}, then Pr{Ai ≤x}=1−Pr{Ai>x}=1−Pr{Ai1>x, …, AiK>x} =1−Πk=1KPr{Aik>x}=1−exp(−(∑k=1Kβik)x)=1−exp(−βix),
(27)
(28)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
i.e., Ai is exponential distributed with parameter βi=Σk=1Kβik. Then For F A (x ) , there is rk
(29) FArk(x)=Pr{Ark≤x}=Σi=1MPr{Ark≤x, r=i}, where Pr{Ark≤x, r=i}=Pr{Aik≤x, Aik≥Aj (j≠i), Aik≤Ait (t≠k)} (30) +Σl≠kPr{Aik≤x, Ail≥Aj (j≠i), Ail≤Ait (t≠l)}. Obviously, the random variables Ai1, …, AiK and Aj’s (j≠i) are independent from each other. Thus for f A , , A , A 's ( j ≠i ) (ai1 , , aiK , a j ' s( j ≠ i )) , the joint probability density i1
iK
REFERENCES
j
function of Ai1, …, AiK, and Aj’s (j≠i), there is f Ai1 ,
Fm(x)=Fm−1(x)(1−exp(−λmx)). (39) Differentiating (39), by recursion and F1’(x)=λ1exp(−λ1x) we have Fm’(x)=Fm−1(x)λmexp(−λmx)+Fm−1’(x)(1−exp(−λmx))=… =λmFm−1(x)exp(−λmx)+λm−1Fm−2(x)exp(−λm−1x)(1−exp(−λmx)) +λm−2Fm−3(x)exp(−λm−2x)(1−exp(−λm−1x))(1−exp(−λmx))+…+ λ1exp(−λ1x)(1−exp(−λ2x))(1−exp(−λ3x))⋅…⋅(1−exp(−λmx)) ■ =∑i=1mλi exp(−λix)Πj≠i (1−exp(−λjx)).
, AiK , A j ' s ( j ≠ i )
K , aiK , a j ' s ( j ≠ i )) = ∏ k =1 f Aik ( aik )∏ j ≠i f A j ( a j ) , (31)
( ai1 ,
note that fΛ(⋅) denotes the PDF of the r.v. Λ. Consequently, since Ai (i=1, …, M) is exponential distributed with parameter βi=∑k=1Kβik (see (28)), according to (31), for the first adding part of Pr{Ark≤x, r=i}in (30), there is Pr{ Aik ≤ x, Aik ≥ A j ( j ≠ i ), Aik ≤ Ait (t ≠ k )}
{∏k '=1 f Aik ' (aik ' )∏ j ≠ i f A j ( a j )}dV K
=∫
Aik ≤ x , Aik ≥ A j ( j ≠ i ), Aik ≤ Ait ( t ≠ k ) x
= ∫ (∏ k '≠ k ∫ 0
+∞
a ik
f Aik ' ( aik ' )daik ' )(∏ j ≠ i ∫
aik
f A j (a j )da j ) f Aik ( aik ) daik
0
x
= ∫ e −( ∑k '≠k β ik ' ) aik (∏ j ≠i (1 − e
− β j a ik
0
x
= ∫ β ik e − β iφ ∏ j ≠ i (1 − e
− β jφ
0
(32)
[1] [2] [3] [4] [5]
)) β ik e − βik aik daik
[6]
)dφ ;
for each sub-adding part of the second part of (30), there is [7]
Pr{ Aik ≤ x, Ail ≥ A j ( j ≠ i, l ≠ k ), Ail ≤ Ait (t ≠ l )} =∫
Aik ≤ x , Ail ≥ A j ( j ≠i ,l ≠ k ), Ail ≤ Ait ( t ≠ l ) x
aik
+∞
0
0
ail
= ∫ ( ∫ (∏t ≠l ,k ∫ x
a ik
0
0
∏
K k '=1
f Aik ' ( aik ' )∏ j ≠i f A j ( a j )dV
f Ait (ait )dait )(∏ j ≠i ∫
ail
0
= ∫ {∫ (∏t ≠l ,k e −βit ail )(∏ j ≠i (1 − e
− β j a il
f A j ( a j )da j ) f Ail ( ail ) dail ) f Aik (aik )daik
))β il e − βil ail dail }β ik e − βik aik daik
φ x β )θ −( −β θ = ∫ {∫ βil e ∑t≠ k it ∏ j ≠i (1 − e j )dθ }β ik e − βikφ dφ . 0
[9] [10]
0
(33)
Substituting (32) and (33) into (30), there is x
Pr{ Ark ≤ x, r = i} = ∫ β ik e − β iφ ∏ j ≠ i (1 − e 0
− β jφ
0
x
(34)
0
= ∫ β ik e − β iφ ∏ j ≠i (1 − e 0
[11]
)dφ
φ x β )θ −( −β θ + ∑l ≠ k ∫ {∫ β il e ∑t≠k it ∏ j ≠i (1 − e j )dθ }β ik e − βik φ dφ − β jφ
)dφ
x φ −( β it )θ −β θ + ∫ {∫ {(∑l ≠ k β il )e ∑t ≠k ∏ j ≠i (1 − e j )}dθ }β ik e −β ik φ dφ . 0
[8]
[12] [13]
0
Substituting (34) into (29), then we obtain the CDF of Ark as shown in (25). Calculating the derivative of FArk(x), we arrive at the PDF of Ark as shown in (26). For FAr(x), the CDF of Ar, there is FAr(x)=Pr(Ar≤x)=Pr(A1≤x, …, AM≤x)=Πi=1MPr(Ai≤x). (35) Substituting (28) into (35), we have FAr(x) as shown in (27). ■ Lemma 2 For random variables X and Y which satisfy X≤Y, there is Pr(Y≤t)≤Pr(X≤t). (36) Proof: Since X≤Y, let Y=X+δ, δ≥0, we have ■ Pr(Y≤t)=Pr(X+δ≤t)=Pr(X≤t−δ)≤Pr(X≤t). Lemma 3 For a function with the form of Fm(x)=Πi=1m(1−exp(−λix)), (37) its derivative function with respect to x is of the form Fm’(x)=∑i=1mλi exp(−λix)Πj≠i (1−exp(−λjx)). (38) Proof: Obviously there is
[14] [15] [16] [17]
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