Outage Probability at Arbitrary SNR with Cooperative Diversity

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					700                                                                                     IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005

               Outage Probability at Arbitrary SNR with
                        Cooperative Diversity
                                              Yi Zhao, Raviraj Adve, and Teng Joon Lim

   Abstract— Cooperative diversity improves the performance of                nodes. Denote the set of all cooperating nodes as {c}. To guar-
wireless networks by having several nodes transmit the same                   antee orthogonal transmissions, we consider a Time Division
information. We present an outage probability analysis for a                  Multiple Access (TDMA) arrangement with m + 1 time slots.
decode-and-forward system, valid at all signal to noise ratios
(SNR). A closed form solution is obtained for independent                     The first slot is used for the source to transmit its signal to
and identically distributed (i.i.d.) channels, and two tight lower            the destination as well as share it with the cooperative nodes.
bounds are presented for correlated channels.                                    If the channel between the source and a node is good
   Index Terms— Outage probability, cooperative diversity, wire-              enough, this node becomes an active cooperating node, and
less cooperative networks.                                                    decodes and forwards the source information. Denote the set
                                                                              of active cooperative nodes as C ⊂ {c}. In the following m
                                                                              slots, the active cooperative nodes repeat the source message
                         I. I NTRODUCTION
                                                                              in a predetermined order [1].
      OOPERATION helps create spatial diversity in wireless
C     networks, even if individual nodes do not use antenna
arrays for transmission or reception [1], [2]. While asymptotic
                                                                                 Assuming that the destination d has exact channel state
                                                                              information (CSI), maximum-likelihood combining of the
                                                                              signals received from all |C| + 1 nodes can be employed. Fur-
(in SNR) performance analysis highlights the diversity order                  thermore, we assume that both the source and the cooperative
achievable by various techniques, it is also important to study               nodes do not have access to any transmit channel information.
performance in the non-asymptotic or finite SNR regime
so as to compare various schemes in practical settings. In
                                                                                III. O UTAGE P ROBABILITY: F ORMULAS AND B OUNDS
this letter, we derive the exact outage probability of the
decode-and-forward scheme1 for arbitrary signal-to-noise ratio                A. Outage Probability for i.i.d. Channels
(SNR) by using two powerful mathematical tools: Moment                          The mutual information between the source and cooperative
Generating Function (MGF) and Order Statistics. We derive an                  nodes c = 1, . . . , m is [4]:
exact closed-form expression for i.i.d. channels, and introduce                                       1
simple lower bounds for non-i.i.d. channels. Simulation results                               Ic =       log (1 + SNRˆc ) ,
                                                                                                                     x                     (1)
show that those bounds are very tight.
   As an example of the utility of our results, consider that                 where xc = |hs,c |2 and hs,c is the complex channel gain
Laneman and Wornell proved in [1] that both the decode-and-                   between source s and node c, modelled as a zero-mean circu-
forward and the space-time-coded cooperation can provide                                                                  ˆ
                                                                              larly symmetric Gaussian random variable. xc is exponentially
“full diversity” in the sense that the diversity order (using                                             ˆ
                                                                              distributed with parameter λc . SNR is the transmit signal-to-
outage probability as a performance measure) is the total                     noise ratio, and the factor m+1 captures the TDMA nature
number of cooperating nodes in the network. However, our                      of the scheme, in which node s is allowed to transmit its
results will show that at finite SNR levels, a higher diversity                information only a fraction 1/(m + 1) of the time.
order does not necessarily translate into better performance.                    If the instantaneous mutual information Ic is higher than the
This implies that the optimal number of cooperating nodes                     transmission rate R, we can assume that the cooperative node
is in fact a complex function of the operating SNR and the                    successfully decodes the source bits, and thus belongs to the
cooperative diversity scheme in use.                                          active cooperative set C, or C = {c : Ic > R, c = 1, . . . , m}.
                                                                                 As presented in [1], the mutual information of the decode-
                        II. S YSTEM M ODEL                                    and-forward transmission is
   We consider a system with a source node, s, communicating                                 1
                                                                                      I=        log 1 + SNR x0 +                xc   ,     (2)
with a destination, d, with the help of m other cooperating                                 m+1
(or relay) nodes, which are called cooperative nodes or relay
                                                                              where x0 = |hs,d |2 and xc = |hc,d |2 are exponentially
   Manuscript received January 18, 2005. The associate editor coordinating    distributed with parameter λ0 and λc , respectively.
the review of this letter and approving it for publication was Prof. Carla-
Fabiana Chiasserini.                                                             The outage probability is defined as Pout = P I < R
   The authors are with the Department of Electrical and Computer En-         where R is the required transmission rate for source s. Using
gineering, of Toronto, Toronto, Ontario, Canada (e-mail: {zhaoyi, rsadve,     the total probability law, we can write the outage probability
   Digital Object Identifier 10.1109/LCOMM.2005.08025.                         as:
   1 Where the cooperative nodes attempt to decode the source’s bits, and                  P I<R =          P I<R|C P C .                (3)
retransmit those bits if the decoding succeeds [3],                                                      C
                                                           1089-7798/05$20.00 c 2005 IEEE
ZHAO et al.: OUTAGE PROBABILITY AT ARBITRARY SNR WITH COOPERATIVE DIVERSITY                                                                                              701

   Generally, this probability is very difficult to compute since                      B. Convenient Lower Bounds
the summation is over all the possible active cooperative sets,
which has 2m items. However, if we assume independent and                                In Theorem 1 we derive the exact outage probability for-
identically distributed (i.i.d.) fading, the closed-form outage                       mula for i.i.d. fading. Unfortunately, Pout does not have a nice
probability can be derived.                                                           and simple form for general non-i.i.d. cases. However, we can
   Theorem 1: Under the assumption that the Rayleigh fading                           still obtain convenient bounds to avoid the complex numerical
from the source to the cooperative nodes are i.i.d. (λc = ˆ                           summation in (3). We present two convenient lower bounds
ˆ c = 1, . . . , m), and those from all nodes to the destination
λ,                                                                                    as the following theorems.
are also i.i.d. (λ0 = λc = λ, c = 1, . . . , m), the outage                             Theorem 2: One lower bound for the outage probability of
probability of the system is                                                          a decode-and-forward system is
               m                       k                          m−k
                     m         ˆ                          ˆ                                               m
      Pout =                e−λγ           1 − e−λγ                                                                             ˆ
                     k                                                                Pout     ≥                  1 − e−λc γ              1 − e−λ0 γ +                        (11)
                                                          (λγ)i                                                   m                                m
                               1 − e−λγ                         ,               (4)                    1−                 1 − e−λc γ
                                                                                                                                                           1 − e−λc γ/(m+1)      .
                                                                                                              c=1                                c=0
            2(m+1)R −1
where γ =      SN R    .                                                                     Proof: Based on the fact that
     Proof: First consider the conditional probability
                                                                                             x0 +         xc ≤ (m + 1)xmax , xmax = max xc ,
                                                k                                                                                                           c=0,...,m
                                                                  2        −1                       c∈C
 P I < R | C = {1, . . . , k} = P                    xc <
                                                                       SN R           first we have
                                                                                      Pout = P x0 +                   xc < γ ≥ P x0 < γ P k = 0 +
   We now use Moment Generating Function (MGF) to find                                                         c∈C
the distribution of xsum = c=0 xc . Since each xc is expo-                                                                 P (m + 1)xmax < γ P k = 0 , (12)
nentially distributed with parameter λ, its MGF is Mc (s) =
  λ                                                                                   where k again represents the size of the active cooperative set,
s+λ . Furthermore, the xc ’s are independent, so the MGF of
xsum is [5]                                                                           or k = |C|.
                                   λ                                                    Using a result in order statistics, we can obtain the CDF of
                  Msum (s) =                .            (6)
                                 s+λ                                                  xmax as [6]
                                                                                                                          m                 m
  Applying the inverse Laplace Transform, we can get the
                                                                                                    Fmax (x) =                  Fc (x) =           1 − e−λc x ,
pdf, and then the CDF of xsum as
                                                                                                                          c=0              c=0
                                                k                 i
                                                         (λx)                         and therefore
               Fsum (x) = 1 − e−λx                            ,                 (7)                                                 m
                                                                                          P (m + 1)xmax < γ =                              1 − e−λc γ/(m+1) .           (13)
therefore (5) becomes                                                                                                               c=0
                                                         (λγ)i                          For the other items in (12), it is simple to obtain
               Fsum (γ) = 1 − e−λγ                             .                (8)
                                                           i!                                                                       m
                                               i=0                                                                                                     ˆ
                                                                                                      P k=0                   =            1 − e−λc γ                   (14)
  Notice that this conditional distribution is only determined                                                                      c=1
by k, the size of the active set, and not by the identity of the                                                                                            ˆ
k nodes in the set. As a result, we can re-write (3) as                                               P k=0                   =     1−           1 − e−λc γ             (15)
                     m                                                                                                                     −λ0 γ
                                                                                                     P x0 < γ                 =     1−e            .                    (16)
      P I<R =              P I < R | |C| = k P |C| = k                          (9)
                                                                                         Substituting (13), (14), (15) and (16) into (12) completes
  From the mutual information formula (1), we have                                    the proof.
                                                              ˆ                         Theorem 3: Another lower bound for the outage probability
               P c ∈ C = P xc > γ = e−λγ ,
                                                                                      of a decode-and-forward system is
       ˆ                                             ˆ
since xc is exponentially distributed with parameter λ. It is                                                 m
straightforward then to arrive at                                                      Pout     ≥                 1 − e−λc γ            1 − e−λ0 γ +                    (17)
                                           k                          m−k                                 c=1
                       m           ˆ                          ˆ
     P |C| = k =              e−λγ              1 − e−λγ                       (10)                                   m                          m
                       k                                                                                                            ˆ
                                                                                                          1−               1 − e−λc γ                  ac 1 − e−λc γ     ,
                                                                                                                  c=1                           c=0
  Finally, substituting (8) and (10) into (9), we get (4), which
completes the proof.                                                                  where ac =          i=c λi −λc .
702                                                                                                                                                   IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005

                             0                                                                                                              0
                           10                                                                                                              10

                             −2                                                                                                             −2
                           10                                                                                                              10
                                        m=0                                                                                                                                                               m=7
                             −4                                                                                                             −4
                           10                                                                                                              10
      Outage Probability

                                                                                                                      outage probability
                           10                                                                                                               −6

                           10                                                                                                               −8

                           10                                                                                                               −10

                                                                                                                                                          numerical result
                                                                                                                                                          lower bound 1
                           10                                                                                                               −12           lower bound 2

                           10                                                                                                               −14
                                   0          5       10       15          20       25         30     35   40                              10
                                                                                                                                                  0       5          10         15      20      25   30         35   40
                                                                        SNR (dB)                                                                                                     SNR (dB)

Fig. 1.                         Outage probability for the i.i.d. cases.                                          Fig. 2.                        Outage probability and lower bounds for the non-i.i.d. cases.

                           Proof: Similar to (12), we have                                                           Plots like Fig. 1 help in system design not only because
                                                                                   m                              outage probability is an important QoS parameter in itself, but
   Pout ≥ P x0 < γ P k = 0 + P                                                           xc < γ P k = 0 ,         also because they allow us to determine the optimal number of
                                                                                   c=0                            cooperative nodes. For example, from Fig. 1 we can see that
                                                                                                           (18)   when SNR= 20 dB, the optimal number of cooperative nodes
   The MGF of                                        c=0   xc can be written as                                   is m = 2. When SNR increases to 30 dB, the optimal size
                                                               m                         m                        changes to m = 4. An asymptotic analysis based only upon
                                                                     λc                         ai λc
                                       Msum (s)            =              =                           ,    (19)   diversity order, on the other hand, will always point to m = 8
                                                                   s + λc                c=0
                                                                                               s + λc
                                                                                                                  as the best setting because it yields the largest diversity order.
                                                                        λi                                           Fig. 2 compares the two lower bounds with the actual Pout
                                       where ac            =
                                                                     λi − λc                                      obtained from numerical computations. In this simulation we
                                                                                                                  set R = 1 bit/sec/Hz, λc , λ0 and λc are i.i.d. uniformly
   Therefore the CDF of                                             c=0   xc is                                   distributed in [0, 2]. From the figure we can see that the two
                                                                    m                                             bounds are almost identical and very tight, providing very
                                                  Fsum (x) =            ac 1 − e−λc x                      (20)   good approximations to the outage probability.

  Finally, substituting (20), (14), (15) and (16) into (18)                                                                                                                  R EFERENCES
completes the proof.                                                                                              [1] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded pro-
  Since x0 + c∈C xc ≤          c=0 xc ≤ (m + 1)xmax , the                                                             tocols for exploiting cooperative diversity in wireless networks,” IEEE
second lower bound is tighter, but requires slightly heavier                                                          Trans. Inform. Theory, vol. 49, pp. 2415–2425, Nov. 2003.
                                                                                                                  [2] H. El Gamal and D. Aktas, “Distributed space-time filtering for cooper-
computation load.                                                                                                     ative wireless networks,” Proc. IEEE Globecom, pp. 1826–1830, 2003.
                                                                                                                  [3] A. Sendonaris, E. Erkip, and B. Aazhang, “Increasing uplink capacity via
                                                   IV. S IMULATION R ESULTS                                           user cooperative diversity,” Proc. IEEE Int. Symp. Info. Theory, p. 156,
                                                                                                                      August 2002.
  Fig. 1 shows the outage probability of a decode-and-forward                                                     [4] T. M. Cover and J. A. Thomas, Element of Information Theory.
system with the number of cooperative nodes ranging from 0                                                        [5] K. Knight, Mathematical Statistics. Chapman and Hall, 1999.
                                                                                                                  [6] H. A. David and H. N. Nagaraja, Order Statistics: Third Edition. John
to 8. In the simulation we set R = 1bit/sec/Hz, and λc =ˆ                                                             Wiley, 2003.
λ0 = λc = 1, c = 1, . . . , m.

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