where L is the number of columns in the space-time code matrix, and N is the number of relaying nodes. 1. INTRODUCTION. Using diversity in time, frequency, ...
RANDOMIZED DISTRIBUTED SPACE-TIME CODING FOR COOPERATIVE COMMUNICATION IN SELF ORGANIZED NETWORKS Birsen Sirkeci-Mergen and Anna Scaglione School of Electrical & Computer Engineering, Cornell University, Ithaca, NY 14853 ABSTRACT We consider a wireless network in which a single source transmits its message with the help of multiple cooperative relays to a far away destination. In our model the source broadcasts the message and only the relays that are able to produce an inference of acceptable quality retransmit the message by using orthogonal spacetime codes. The novelty of our transmission protocol is that, unlike other works on cooperative space-time coding, the code assignment is performed in a completely distributed way by having each node pick one column of the space-time code matrix at random. The interesting result of our analysis is that this simple method achieves the maximum possible diversity order for signal-to-noise ratio (SNR) below a threshold SNRt , which increases with node density. Hence depending on the SNR range of a given system, one can achieve the maximum diversity order by increasing the number of relay nodes. In addition, for any SNR, the average error probability of the scheme decreases proportional to LN , where L is the number of columns in the space-time code matrix, and N is the number of relaying nodes. 1. INTRODUCTION Using diversity in time, frequency, space or any combination of them can improve the performance of wireless systems in the presence of fading. For systems with multiple antennas space-time (ST) codes are designed to achieve the maximum diversity order at minimum possible loss in rate. In order to have ST coding gains, the hardware complexity is increased (not the bandwidth) because of the need of a separate RF chain to support the multiple antennas transmission and reception. In ad-hoc networks applications or in distributed large scale wireless networks, the nodes are often constrained in the complexity of their hardware and also in their size. The latter constraint prevents using antennas that are sufficiently far apart to provide independently faded links with the destinations. In addition, the hardware complexity is excessive. This makes the classical space-time codes impractical for certain network applications. For such cases, recent works have shown that cooperation among the transmitters can provide spatial diversity, which is also called cooperative diversity (see e.g. [1, 2]). The space-time codes in cooperative networks are called distributed ST codes since the antenna array is distributed in space. However, most of the so called distributed space-time codes proposed for cooperative networks are not really result of a distributed protocol and require some form of central control that assigns the space-time code matrix columns to the cooperating nodes [3–6]. This work is supported by National Science Foundation under grant ITR - 0428427.
In this paper, our contribution is two fold: (i) we propose a new, truly distributed space-time coding scheme for cooperative networks, (ii) we analyze the proposed protocol. In the cooperative network, we assume that there is a single source-destination pair and multiple relay nodes. When the destination is remote with respect to the network, this problem is often referred to as the reach-back problem (see Fig. 1). The source node transmits the message to the relaying nodes. After the source transmission, the relay nodes use an orthogonal ST code for the retransmission of the message to the destination. In our protocol, the assignment of the space-time code matrix columns among the relays is done randomly. More specifically, if the ST code is an L × L matrix, each node will pick any of the columns with equal probability L1 . We choose orthogonal ST codes because they have the advantage of allowing simple decoding at the destination; furthermore they are appropriate when the number of transmitters is unknown, as is the case in cooperative networks. This issue has also been mentioned in [6, 7] previously. Our main observation in this work is that the proposed scheme achieves maximum possible diversity order for signal-to-noise ratio (SNR) less than a threshold SNRt . The threshold SNRt increases with the node density; hence depending on the SNR range of a given system, one can achieve the maximum diversity order by increasing the number of relay nodes. For finite number of relaying nodes and SNR > SNRt , the diversity order is 1 because there is some finite probability that all nodes will choose the same code. However, we show that the probability of error decreases asymptotically (high SNR) like O( L1N ), where N is the number of relaying nodes and L is the number of columns in the code matrix.
Fig. 1. Reach-back problem The paper is organized as follows: In Section 2, we describe the system model. In Section 3 we explain the proposed distributed space-time coding scheme. In Section 4, we derive the exact prob-
ability of error at the destination node, and the asymptotic approximations are derived in Section 5. In Section 6, we present the simulations. 2. SYSTEM MODEL We consider a cooperative network formed by a single sourcedestination pair and multiple relaying nodes that facilitates the source transmission to a remote destination. In our protocol, the source node initiates the transmission session by sending a packet encoded at a rate R < log(1 + γ), where γ is a given SNR threshold. Let s0 s1 . . . sn be the sequence of symbols that are transmitted. We assume that the channel between any two nodes is affected by a random small scale flat fading, a deterministic large scale fading, and additive white Gaussian noise (AWGN). For simplicity, we neglect the propagation delays. The nodes that have received the source message with SN R ≥ γ are responsible for relaying the message to the destination, assuming that the packet is long enough to achieve almost reliable reception at the relay nodes whose receiver SN R ≥ γ. Under these conditions only the nodes that do not have an outage retransmit. Note that the message has a known preamble that the receiving node use to estimate the SNR and synchronize their retransmission. Let T be the set of indices of the relay nodes, and define the cardinality of the set as |T |. The source and the destination Swill be denoted by the letters, c and d respectively. Let i, k ∈ T {c, d}. We denote the channel gain between the i’th and k’th relay node as hik which is equal to: hik =
αik , dβik
the receiver, has been previously discussed in [8]. OSTC has also been used in [6] for the same practical reason. Let S ⊂ T be the set of relays that are capable of decoding the source message correctly. And, let Si , i = 1 . . . L be the set of indices of the S nodes that chose the i’th column of the code matrix G; clearly L i=1 Si = S. Since T a node can only select a single virtual antenna, for i 6= j, Si Sj = ∅. Then the received signal at the destination is r = Gh + n, where n ∼ CN (0, N0 I) is AWGN noise at the destination, h = [h1 h2 . . . hL ]t such that hi =
N X
3. RANDOMLY DISTRIBUTED SPACE-TIME CODES During retransmission, the relaying nodes uses a space-time code with a given dimension, which is denoted by L. The nodes synchronize their transmission through the message preamble and are programmed to retransmit at unison after a suitable interval of time elapsed since the estimated beginning of the source message. Unlike [6], given the space-time code matrix, the assignment of the code matrix columns among the relaying nodes is done in a distributed fashion. That is, each node chooses randomly to serve as one of the L virtual antennas independently (this can be imagined as each node using a fair L dimension die for the virtual antenna selection). A space-time block code is represented by a p × L matrix G, where L is the number of transmission antennas, and p is the number of time slots used during transmission. The elements of the code matrix G are linear combinations of the source symbols s1 , . . . sn and their complex conjugates. The rate of the code is defined as R = np . Note that the set of relaying nodes is the random set of nodes whose SN R ≥ γ; as a result, the number of nodes in this set is random. This puts a limitation on the choice of space-time codes to be used in such networks. We choose the code matrix to be an orthogonal space time code (OSTC) matrix. The issue of spacetime code design, when the number of transmitters are unknown at
(2)
where 1A is the indicator function. 4. CALCULATION OF EXACT AVERAGE PROBABILITY OF ERROR AT THE DESTINATION The performance analysis of OSTC can be simplified using the channel decoupling property [9]. That is, the ML detection for OSTC can be decomposed into n scalar detection problems of unknown symbol si , i = 1 . . . n in AWGN. Let’s assume each relay transmits with power P Pr . 2Note that the OSTC code matrix G satisfies, G H G = n i=1 |si | I. Then the SNR on the n’th complex channel is equal to
(1)
where dik denotes the distance between i’th and k’th node and αik is circular complex Gaussian with zero mean and variance 1 (i.e. Rayleigh fading). We assume that the fading is constant during the entire transmission of the message.
hkd 1{k∈Si } ,
k=1
SN Rn =
Pr |h|2 E[|sn |2 ]. N0
(3)
For details refer to [9]. The probability density function f|h|2 (x|S1 , . . . , SL ) of |h|2 given (S1 , . . . , SL ) can be easily calculated since (hi |S, Si ) ∼ CN (0, σi2 ), where σi2 (S, Si ) =
N X 1 k=1
d2β kd
1{k∈Si } .
The hi , i = 1 . . . L are conditionally independent since hkd ’s for k ∈ T are independent. In the following we will assume σi2 6= σj2 for i 6= j except the case when both are zero. This assumption is reasonable since two nodes deployed at random will have equal distance from a common point with probability zero. Note that the degenerate case σi2 = 0 represents hi = 0, which is the case when none of relays chooses to serve as the i’th antenna. S
T
Lemma 1 Let Si ⊂ S such that i Si = S. Also, Si Sj = ∅ for i 6= j. Assume that nonzero σi2 ’s are distinct. The density function of |h|2 conditioned on S1 , . . . SL is as follows: f|h|2 (x|S1 , . . . SL ) =
L X i=1
"
#
(σ 2 )L−2 x Qi exp(− 2 ), σi j6=i fij
where
fij =
1 σi2 − σj2
if σi2 = σj2 = 0 otherwise.
Proof The proof follows easily from (2), since hkd are normal distributed, the calculation of the characteristic function for |h|2 conditioned on S1 , . . . SL and making a partial fraction expansion is sufficient. Note that σi2 depends on the random sets Si , S and fij depends on the random sets Si , Sj , S. Next we will average over all possible combinations of S1 , . . . , SL that satisfy the conditions given in lemma 1. There are L|S| possible such combinations. Let Pm = [Pm1 . . . PmL ] denote the m’th possible combination such that Si = Pmi in the m’th combination for m = 1 . . . L|S| . Then, the distribution of |h|2 given S is p(x|S) =
X
P r(S1 = Pm1 , . . . , SL = PmL )
Pm
×p(x|Pm1 , . . . PmL )
" # L XX (σk2 (Pmk ))L−2 /L|S| −x/σk2 (Pmk ) Q = e j6=k
Pm k=1
(4)
fkj (Pm )
We now assume that the space time code is constructed with BPSK equally probable symbols ±1. Then the error probability p Pr conditioned on |h|2 is Q( 2Pr |h|2 SN Rr ), where SN Rr , N 0 R∞
where P(T ) denotes the power set of T . Remark: If the protocol is updated in such a way that the source also joins the relaying nodes in retransmitting the message, the above probabilitySof error expression can be adjusted easily by replacing T with T {c}. To understand more about the average error probability, in the next section we look at the behavior of (6) at the asymptotic regimes. 5. APPROXIMATION FOR AVERAGE PROBABILITY OF ERROR IN ASYMPTOTIC REGIMES In this section, we look at the behavior of average probability of error under two different asymptotic regimes. The first case is similar to the conventional diversity analysis, that is the regime when SN R→∞ at the destinations. In the second case, we fix the total relay power and assume the relay node density goes to infinity. In Section 6, we validate the conclusions we derive in this section using simulations and numerical evaluations of (6). Pc Pr Lemma 2 Define SN Rs = N and SN Rr = N . Let T be 0 0 the finite set of relay nodes. As N0 →0, i.e. SN Rs →∞ and SN Rr →∞, the average probability of error converges to
2
and Q(a) = a √12π e−u /2 du. Hence, the probability of error pe conditioned on the set S, using (4) is; Z
pe |S
∞
=
√ Q( 2xSN Rr )p(x|S)dx
where σ 2 =
0
=
" # L XX (σk2 (Pmk ))L−2 /L|S| Q Pm k=1
Z
j6=k ∞
×
fkj (Pm )
√ 2 Q( 2xSN Rr )e−x/σk (Pmk ) dx
0
Using the result
R∞ 0
2
xe−x
/2
Q
x σ
1 2
dx =
1− √
1 σ 2 +1
,
we can simplify the expression for average probability of error; # " L XX (σk2 (Pmk ))L−2 /L|S| Q g(σk2 (Pmk )) pe |S = j6=k fkj (Pm )
Pm k=1
where σ2 g(σ ) = 2 2
r
1−
σ 2 SN Rr σ 2 SN Rr + 1
(5)
!
.
In order to find pe , we need to average the pe |S over the possible set of relays. Let’s assume the source transmits with power Pc , and the channel gains between the source and the k’th relay is denoted by hck = αck /dβck . After the source transmission, the 2
ck | > τ retransmit, nodes with sufficient SNR, i.e. with Pc |h N0 2 where τ is a given threshold. Since |hck | is exponentially distributed, the probability that the k’th node joins the set S is given by, exp(− τPNc0 d2β ck ) and the average probability of error is given by,
pe =
" # L X X (σk2 (Pmk ))L−2 /L|S| Q g(σk2 (Pmk ))
X
S∈P(T ) Pm (S) k=1
×
Y i∈S
e
−
j6=k fkj (Pm )
! 2β
τ N0 d ci Pc
Y i∈S /
pe →
−
(1 − e
2β τ N0 d ci Pc
!
)
P k∈T
(7)
−2β dkd .
Proof As SN Rs →∞, every relay node in T has sufficient SNR to decode the source message correctly, and join the retransmission session. As SN Rr →∞, the dominating term in the pe expression given by (6) corresponds to the case where Si = T , Sj = ∅, ∀j 6= i, for i = 1 . . . L. These are the L cases such that each relay picks a specific column of ST code matrix. Each node has probability L1 of picking a specific column, and the probability that all nodes pick the same column is ( L1 )|T | . Also, the function g(σ 2 )→ 4SN1 Rr as SN Rr →∞. Hence, the result in (7) follows. Notice the dependence of σ 2 on T . Using (7) suggests that the diversity order of the proposed scheme is 1, which somewhat misleading. In fact, for large N , |T | values (for example N > 10) the multiplicative term of SN1Rr in (7) is small enough that the diversity order 1 dominates at extremely high SNRs that have no practical interest. For this reason, we look at another asymptotic regime where number of relaying nodes is infinite. Lemma 3 Let the active relay nodes (responsible for retransmitting) be distributed randomly in a given area A and assume the destination is located at (xd , yd ). We assume that total relay power is fixed, i.e. the relay power density P¯r , PrAN is fixed. Assume that nodes uses equal probable BPSK at ±1. Given finite SN Rr , as N →∞, pe → where µ =
m L−1 (1 − µ)L X 1+µ L−1+m , m 2L 2 m=0
q
SN R 1+SN R
SN R = (6)
L(1−|T |) , 4σ 2 SN Rr
(8)
and
Z 1 1 P¯r dudv N0 L A ((xd − u)2 + (yd − v)2 )β
and L is number of columns in the ST code matrix.
(9)
Proof Since the total relay power is fixed, the SNR at the n’th complex channel (3) is scaled such that P¯r A |h|2 E[|sn |2 ] SN Rn = . N N0 Define Xk = [hkd 1{k∈S1 } , hkd 1{k∈S2 } , . . . , hkd 1{k∈SL } ]t for k = 1 . . . N . We can rewrite h in terms of the random vectors Xk , i.e. h = [h1 h2 . . . hL ]t =
N X
6. SIMULATIONS In this section we check the accuracy of the average error probability expression derived in Section 4 and also get some intuition about the system behavior with respect the network parameters such as number of relay nodes and number of columns in the space-time code matrix G. node locations 0.3
Xk
0.2 0.1
k=1
(a)
where hi is given by (2). In the following, first we derive the mean and variance of Xk and then by using the multivariate central limit theorem as N →∞, we show that √hN converges in distribution to a Gaussian random variable. We know that the nodes are randomly distributed and each node chooses independently to serve as the i’th antenna with probability L1 . Then the mean of Xk is
E{Xk } = 0,
−0.2
−0.4 −1
{i=j}
(10)
which follows from the independence of hkd and 1{k∈Si } , ∀i. Since the nodes are uniformly distributed, Σ = σ 2 I where σ2
, L1 A1
Z A
0
0.5
1
1.5
2
Pe versus SNR (dB)
0
10
simulation numerical evaluation
−2
10 −10
−8
1 dudv. ((xd − u)2 + (yd − v)2 )β
−4
−2
0
2
4
6
8
10
(11)
In the following we deal with a network formed by a single source/destination pair and N relay nodes such that the relays are uniformly distributed in this disc with radius R = 1. The topology of the network is fixed. Furthermore, the simulations and the analysis is based on the nodes using BPSK modulation. In Fig. 2b, we validate the accuracy of the probability of error expression (6) for a network with two relay nodes (the location of the nodes are given in Fig. 2a). The probability of error is averaged over random channel coefficients, random selection of virtual antennas and the noise through 1000 trials. The numerical evaluation of the expression (6) is also given in Fig. 2b and it matches the simulation results.
Note that Σ does not depend on k. Since Xk ’s are independent random vectors and Xk ’s are identically distributed, using the multivariate central limit theorem, we can conclude that as N →∞ (12)
where σ 2 is given by (11). 2
Due to (12), |h| converges in distribution to a χ2 of order N L. Hence, we can easily calculate the asymptotic average error probability as in (8) (see [10] for details). Remark: As N →∞, the asymptotic average probability of error (8) achieves full diversity, that is
Ps = 1, Pr = 0.5, N = 2
0
10
Average probability of error at destination
N 1 X √ Xk →d N (0, σ 2 I) N k=1
−6
Fig. 2. Distributed Alamouti scheme
j
L 1 1 E{ 2β } L 1{i=j} dkd
=
−0.5
j
i
=
R2
−0.3
−1
i
=
D
S
(b) 10
E{|hkd |2 1{k∈S } 1{k∈S } } E{|hkd |2 }E{1{k∈S } 1{k∈S } } E{|hkd |2 } 1
=
0 −0.1
due to the facts that random variables hkd and 1{k∈Si } , are independent and E{hkd } = 0. Let the covariance matrix Σ = E{Xk XH k }. The (i, j)’th term of the covariance Σ is Σij
R1
L=1 L=2 L=3 L=4 −1
10
−2
10
−3
10
!
pe ≈
2L − 1 1 (4SN R)L L
where SN R is given by (9).
−4
10 −10
−5
0
5
10
15
20
25
SNR (dB)
Fig. 3. Average error probability behavior w.r.t. L
30
Ps = 1, Pr = 1/N, L = 2
0
10
Average probability of error at destination
In Fig. 3, we look at the behavior of average probability of error with respect to the number of columns in the space-time code matrix, or the number of available virtual antennas L. For the network given in Fig. 2a, the relays uses a space-time code of dimension L = 2, 3, 4. Note that L = 1 refers to the case where every node if active retransmits the same code. As expected, the average error probability decreases as L increase; furthermore from Lemma 2, we know that as SN R→∞, the average error probability decreases proportional to L1N . In the next experiment, we use a network of ten nodes; the node locations are shown in Fig. 4. In order to see the behavior of the average probability error with respect to the number of nodes in the network N , we fix the total relay power and take the average of the error probability expression (6) over the possible node combinations. For example, for N = 4, each relay transmits with 10! power 1/N = 0.25 and there are 4!6! = 210 such possible combinations we take average over. The numerical evaluation of (6) for N = 2, 4, 6, 10 is displayed besides the asymptotic result (8) in Fig. 5. The number of virtual antennas are fixed to L = 2.
N=2 N=4 N=6 N = 10 N=∞
−2
10
−4
10
−6
10
−8
10
−10
10
−12
10
−10
0
10
20
30
40
50
60
70
SNR (dB)
Fig. 5. Average error probability behavior w.r.t. N
0.8
[3] P. Anghel, G. Leus, and M. Kaveh, “Distributed Space-Time Coding in Cooperative Networks,” in Proc. of 5th NORDIC signal processing symposium, 2002.
0.6
0.4
[4] S. Barbarossa and G. Scutari, “Distributed space-time coding strategies for wideband multihop networks: regenerative vs. non-regenerative relays,” in Proc. of ICASSP, 2004.
y
0.2
0
[5] S. Barbarossa and G. Scutari, “Distributed space-time coding for multihop networks,” in Proc. of IEEE International Conference on Communications, June 2004, vol.2, pp. 916-920.
−0.2
−0.4
[6] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” in IEEE Trans. Inform. Theory, vol. 59, no. 10, Oct. 2003.
relay nodes −0.6
destination source
−0.8 −1
−0.5
0
0.5
1
1.5
2
x
Fig. 4. Network As expected, the curves have a knee, which becomes more pronounced as N increases. Beyond a certain SNR, they all have the same slope, but the knee occurs at high SNR. For SNR values less than a threshold, the average error probability decreases as predicted from the expression derived in lemma 3, while it decreases as in the expression derived in lemma 2 for SNR values larger than the threshold. This can be clearly seen for N = 10 which has a breakpoint around SNR = 35dB. 7. REFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation - Part I: System description,” and “User cooperation - Part II: Implementation aspects and performance analysis,” IEEE Transactions on Communications, v. 51 , no. 11, Nov. 2003. [2] J. N. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” in IEEE Transactions on Information Theory, vol.50, no.12, Dec. 2004.
[7] J. Mietzner, R. Thobaben, and P. A. Hoeher, “Analysis of the expected error performance of cooperative wireless networks employing distributed space-time codes,” in Proc. IEEE Global Telecommun. Conf. (Globecom 2004), Dallas, Texas, USA, Nov./Dec. 2004. [8] P. Maurer and V. Tarokh, “Transmit dievrsity when the receiver does not know the number of transmit antennas,” in Proc. of International Symposium on Wireless Personal Multimedia Communications (WPMC), Sep. 2001 [9] E. G. Larsson and P. Stoica, “Space-Time Block Coding for Wireless Communications,” Cambridge University Press, 2003 [10] J. G. Proakis, “Digital Communications,” Third Edition, 1995.
