In this paper, we present a coding strategy for wireless relay networks where the relay nodes are small devices with few resources, while the source and sink are equipped with multiple antennas to increase the transmission rate. We assume no channel knowledge at all, and the receiver decodes knowing none of the channel paths. This coding scheme uses distributed space-time coding techniques and is inspired by noncoherent differential space-time coding. It is shown to yield a diversity linear in the minimum number of transmit/receive antennas times the number of relays

Hassibi, Babak

Oggier, Frédérique

English

Caltech Authors - Main

A CODING SCHEME FOR WIRELESS NETWORKS WITH MULTIPLE ANTENNA NODES
AND NO CHANNEL INFORMATION
Frederique Oggier and Babak Hassibi
Department of Electrical Engineering
California Institute of Technology
Pasadena, CA 91125
{frederique,hassibi}@systems.caltech.edu
ABSTRACT
In this paper, we present a coding strategy for wireless relay
networks where the relay nodes are small devices with few
resources, while the source and sink are equipped with mul-
tiple antennas to increase the transmission rate. We assume
no channel knowledge at all, and the receiver decodes know-
ing none of the channel paths. This coding scheme uses dis-
tributed space-time coding techniques and is inspired by non-
coherent differential space-time coding. It is shown to yield
a diversity linear in the minimum number of transmit/receive
antennas times the number of relays.
Index Terms- channel coding, mimo systems, networks.
1. INTRODUCTION
Coding for wireless networks has recently received a lot of at-
tention. Using the knowledge acquired from space-time cod-
ing, where the transmit antennas cooperate to resist the fad-
ing, so-called cooperative diversity schemes techniques have
been proposed, where the relay nodes form a virtual multiple
antennas array to obtain the diversity advantage known to be
achieved by MIMO systems [9, 1, 5, 13]. These works have
focused on different aspects of wireless networks coding. In
[5], distributed space-time codes (DSTC) are introduced, and
their pairwise probability of error and the diversity gain com-
puted. This work has been generalized in [6], for the case
where the transmitter and receiver nodes both have several
antennas. In [10], a construction of DSTCs for the multiple
antenna case is presented.
All the above work assume some channel knowledge, ei-
ther at the receiver, at the relays, or both. Coding strategies for
the case when no channel information is available have been
proposed independently in [8, 11, 7]. See also [14, 12] for
different approaches to non-coherent DSTCs. All are based
on noncoherent differential MIMO coding.
This work was supported by NSF grant CCR-0133818, by Caltech's Lee
Center for Advanced Networking and by a grant from the David and Lucille
Packard Foundation.
The goal of this work is to propose a noncoherent scheme
for a wireless network where the transmitter and receiver nodes
are equipped with multiple antennas. This generalizes the
approach of [11] to the multiple antenna case. We start by
recalling the multiple antenna nodes network model in Sec-
tion 2. Section 3 presents the generalized differential scheme
adapted for such networks. The diversity result is provided in
Section 4.
2. DISTRIBUTED SPACE-TIME CODING WITH
MULTIPLE ANTENNAS
Consider a wireless network as in [6], with R + 2 randomly
and independently distributed nodes. A source and a sink,
equipped with several antennas, use the R other nodes as re-
lays. Every relay node has few resources and thus a single
antenna. Denote the channels from the M antennas of the
transmitter to the i-th relay as fli, f2i, . . , fmi, and the chan-
nels from the i-th relay to the N antennas at the receiver as
gil, gi2, ... giN (see Fig. 1). All channels are assumed iid
CAf(O, 1). The total power of the system is denoted by T.
It has been shown in [6] that the optimal power allocation
is to have, for one transmission, P1 = T/2 at the transmit-
ter and P2 = T/(2R) at each relay. We also assume a co-
herence interval T during which fmi and gin are constant.
The information bits are thus encoded into T X M matrices
S = [si, . .. , sm], where sm is a T-dimensional signal sent
by the m-th transmit antenna. The signal S is normalized as
E[Tr(StS)] = M. (1)
To send S , we consider the two-step strategy of [5, 6].
Step 1: at the transmitter. From time 1 to T, the transmitter
sends VPT/MS. The received signal at the i-th relay at
time T is denoted by riT. Similarly, ViT denotes the additive
noise at the i-th relay. The received signal at the relays is
ri = P1T1 E fmiSm + vi.
ml
(2)
1-4244-0728-1/07/$20.00 ©2007 IEEE III -413 ICASSP2007
Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on April 3, 2009 at 13:17 from IEEE Xplore.  Restrictions apply.
Fig. 1. The wireless network model with several antennas.
been argued [2] that the codeword S has to be unitary. To
obtain a differential scheme, we follow the approach of [11]:
1. we design the matrices Ai at the relays and an initial
transmitted signal so that S is unitary,
2. we define a differential encoder at the transmitter node.
3.1. Designing the relays and an initial signal
Recall that here
S =[A,S,. ... ,ARS].
Step 2: at the relays. From time T + 1 to 2T, the i-th relay
sends til .... tiT. The received signal and noise at the n-th
antenna at time T + T are denoted resp. by XTn and WTn. The
noises are iid CA\(O, 1). We have at the nth receive antenna
R
Yn ginti + Wi (3)
The idea behind DSTBs is to design the transmit signal at
every relay as a linear function of its received signal:
IP1+
where Ai is a TX T unitary matrix (so that the protocol is eq-
uitable among different users and time instants, and the noise
forwarded by the relays remains white). The normalization
factor is such that the average transmit power is P2T at each
relay (see [6]). If we summarize the channel model, denoting
by fi = [fl, . , fM]QT, we get from (3) and (4) that
flgln R
Yn = CT[AlS ARS] + [TE ginAivi+w,
fRgRn 1
with cT- PI1P2T and c' P2(TP(P+J)M T P=+1
By defining Y = [Y1, Y2,..., YN], S = [A1S A2S ... ARS],
gi [gil, 9i2,... ,giNl, H = [figl,... ,fRgR]T, and
R R
W = cT gil Aivi + wl, ... giNAivi + WNI
we finally get that the channel model can be written as
(Pi +( )M H± (5)
where Y is a T X N matrix, while S is a T X MR matrix.
3. A GENERALIZED DIFFERENTIAL SCHEME
Let us now consider (5) and assume that the fadings are un-
known. In a traditional noncoherent MIMO setting, it has
The aim is to design the signal constellation .S1.... , SL } 3
S and the unitary matrices Ai, i = 1, . . ., R, so that the Tf
MR matrix S is unitary, i.e, StS = IMR.
First, we choose the codewords to be of the form
Sj =UjS',j'=l,..L
where Uj is a unitary T X T matrix, and S' is an initial trans-
mitted signal normalized so that E[Tr(SjSj)] = M for all
j = 1, ... , L. Furthermore, we require that both the unitary
matrices Ai, i = 1, ... , R, and Uj, j = 1, . . . , L are diagonal.
Thus, denoting by Sj = [AlSj, .... ARSj], we have
S1S.j I [A,UjS', ..., ARUJS/lt[A,UjS', ... ., ARUjS'1
[AS/, ..., ARS/lt[A,S/,.. ,ARS/]
and the fact that Sj is unitary only depends on the choice
of the matrices at the relays and the transmitted signal S'.
We thus design the matrices Ai and S' independently of the
codebook of matrices Uj.
Our construction is based on Butson-Hadamard matrices
(eg. [3]). Recall that a Generalized Butson-Hadamard (GBH)
matrix is a T X T matrix M with coefficients in a ring such
that
MM* = M*M = TIT
where M* is the transpose of the matrix of inverse elements
of M: Tn* = Tn* 1. If the coefficients of M are chosen to
be roots of unity, then mTn* =Tmij, i.e., the inverse is the
conjugate, so that
MMt = MtM = TIT.
Note that the tensor product of two GBH matrices is again a
GBH matrix [3].
Suppose now that we have T = MR, with M the num-
ber of transmit antennas, R the number of relays, and T the
coherence time. Choose a first Butson-Hadamard matrix A of
size R X R, and a second one B of size M. Take
Ai = dMag(ali.... ali aRi, * , aRi),
M M
III- 414
Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on April 3, 2009 at 13:17 from IEEE Xplore.  Restrictions apply.
which is unitary since all aij's are roots of unity, and similarly
S'- F
(
bM,1
bM,1
... bl,M
bM,M
... b1,M
bM,M /
where lS'27 is the normalization factor such that E[Tr(S'tS')]
M. By definition of the Frobenius norm,
M T
/JSF = S 2 MT,I:lj 1 ' =V
s i=aj=a
since all coefficients Si j are roots of unity. By construction
Step 2: at the relays. Each relay Ri, i = 1,... R, con-
secutively multiplies its received signal by a unitary matrix
Ai (Ai is built as described above) and then transmit ti (t) =
Airi(t), and ti(t + T) = / Airi(t + T) . The
received signals are similar to (5):
Y(t) = CT[AlSt A2St ... ARSt]H + W(t)
Y(t + T) = CTU(Zt+T)[ASt ... ARSt]H + W(t + T),
where last equation holds since Ai and U(zt+T) commute,
for all i and for all U(zt+T). We can thus plug the equation
for Y(t) into the one of Y(t + T), which yields
(6)
where
W(t, t + T) = &U(Zt+T)W(t) + W(t + T)
TA B,
and
[A1S',S... ,ARS'lt[AS',S.... ,ARS'I =RM IRMT IRM,
since A & B is a GBH matrix of size RM. For example,
choose forM = 2 and R = 3 the following matrices:
1 f32 T and(B 11)I
The unitary matrices at the relays are given by A1 = 16,
A2 = diag(l, 1,T3,T3,T3,T3), A3 = diag(l, 1,T3,T3,T3,T3),
and the initial transmitted signal is
S/ = 1 1 1 1 I I T
11 1 1 )
3.2. A Differential Encoder
The two-step transmission of Section 2 can be adapted to al-
low for differential encoding and decoding as follows. Let
U(Zt+nT) be the unitary matrix that encodes the data Zt+nT
(to be sent at time t + nT).
Step 1: at the transmitter. The transmitter sends consec-
utively St = U(zt)S' and St+T = U(zt+T)St. At the ith
relay, the received signals are
ri (t) = /T MSt fi +vi (t)
ri(t + T) = PT MU(zt+T)Stfi + vj(t + T)
is the noise term, and W(t), W(t + T) are given as in (5).
Note that the channel coefficients fi, gi do not appear in (6),
meaning that one can decode without knowing the channel.
4. DIVERSITY COMPUTATION
Emulating the point to point case, a natural candidate for the
differential decoder is
arg mmin IY(t + T) & UiY(t) l12.Ul I=1I....L
Let us restrict to the case where T = MR. We consider two
instances of the noncoherent channel (5)
Y(t) = VCTS(t)H + W(t)
where H is an T & N matrix unknown at both the transmitter
and receiver. Since S(t) and S(t)T are indistinguishable for
an arbitrary unitary T X T matrix T, we preprocess the signal
so that
Y(t)
( Y(t+T)J CT( Uk )H+ W(t + T) '
for Uk a unitary matrix of the codebook. Because of the two
steps transmission, both the noise W(t) and the channel ma-
trixH contains products of Gaussian random variables, which
makes a precise analysis difficult. Following [11], we con-
sider a mismatched decoder, and claim that such a decoder
actually already gives the diversity. More precisely, we can
prove that the diversity gain of the coding strategy described
above is
dfmin{M,N}R M#7N
d MR(1 loglogP M =N
III - 415
where fi = [f,...... fM,]T, i = 1. ... . R.
9
Y(t + T) = U(Zt+T)Y(t) + W(t. t + T) .
[A,S/. ... . ARS/I
Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on April 3, 2009 at 13:17 from IEEE Xplore.  Restrictions apply.
that the receiver has to decode without knowing the channel
paths during communication. In order to propose a coding
strategy, we generalized the concept of differential distributed
space-time coding, and showed that we expect a diversity that
is linear in the minimum number of antennas at the transmit-
ter/receiver, times the number of relays.
6. REFERENCES
Fig. 2. A 6 nodes network with 1 antenna versus a 3 nodes
network with 2 antennas.
assuming that (Uk X Ul)(Uk X U1)t is full rank (that is, the
code is fully diverse). The proof is omitted for lack of space,
but we briefly outline the main steps. We are interested in
computing the pairwise probability of the above channel, i.e.,
P(Uk , Ul) = Eg[P(Uk -, U1 g)], k 7 1.
The main difficulty compared to the work in [11] is that the
columns of the channel matrix H are not independent any-
more, and the matrix H is not full rank. Thus the way of doing
the computation is to actually vectorize the system, and then
compute the pairwise probability of error knowing g. This
computation relies on the technique that consists of comput-
ing the probability in the complex domain to avoid conver-
gence issue, and then, knowing that the probability is real, in
using the real part of the expression to get an upper bound.
Once this bound on the probability is computed, we rely on
the computation of the expectation on g done in [6, Theorem
3], from which follows the diversity gain.
What the diversity computation tells us is that the diver-
sity is linear in the number of relays times the minimum num-
ber of transmit/receive antennas. This is illustrated in Fig. 2,
where the BLER of a 6 nodes network with one antenna at
transmitter and receiver is compared to a 3 nodes network
with 2 antennas at both transmitter and receiver nodes. Note
that both networks use the same codebook, meaning that the 2
antennas network is operating at half the rate. The codebook
used here is a cyclic code [2]. As shown above, the design of
the codebook is an independent issue.
5. CONCLUSION
We considered the problem of coding over a wireless relay
network, where the transmitter and receiver nodes are equipped
with several antennas to increase the rate of the network. We
furthermore assumed that there is no channel information, and
[1] K. Azarian, H. El Gamal, P. Schniter, "On the achievable
diversity-multiplexing tradeoff in half-duplex coopera-
tive channels,"IEEE Trans. Inf Theory,vol. 51, Dec. 05.
[2] B. Hochwald,W. Sweldens,"Differential unitary space
time modulation", IEEE Trans. Comm,vol.48,Dec.2000.
[3] K. J. Horadam,"A generalised Hadamard transform", in
the proceedings of ISIT2005, Adelaide.
[4] B. Hughes, "Differential space-time modulation," IEEE
Trans. Inform. Theory, vol 46, Nov. 2000.
[5] Y. Jing and B. Hassibi, "Distributed space-time coding
in wireless relay networks," IEEE Trans. on Wireless
Communications, vol. 5, no 12, Dec. 2006.
[6] Y. Jing and B. Hassibi, "Cooperative Diversity in Wire-
less Relay Networks with Multiple-Antenna Nodes",
submitted to IEEE Transactions on Signal Processing.
[7] Y. Jing and H. Jafarkhani, " Distributed Differential
Space-Time Coding for Wireless Relay Networks", sub-
mitted to IEEE Trans. on Communications.
[8] T. Kiran and B. Sundar Rajan, " Partially-coherent dis-
tributed space-time codes with differential encoder and
decoder," in the proceedings of ISIT 2006, Seattle.
[9] J. N. Laneman and G. W. Womell, "Distributed Space-
Time Coded Protocols for Exploiting Cooperative Di-
versity in Wireless Networks," IEEE Trans. Inf Theory,
vol. 49, no 10, Oct. 2003.
[10] F. Oggier, B. Hassibi,"An Algebraic Coding Scheme
for Wireless Relay Networks with Multiple-Antenna
Nodes", subm. to IEEE Trans. Signal Proc., Mar. 2006.
[11] F. Oggier, B. Hassibi,"A Coding Strategy for Wireless
Networks with no Channel Information", AllertonO6.
[12] T. Wang, Y. Yao, G. B. Giannakis, "Non-Coherent Dis-
tributed Space-Time Processing for Multiuser Coopera-
tive Transmissions," Proc. of Globecom Conf. 2005.
[13] S. Yang and J.-C. Belfiore, "Optimal Space-Time Codes
for the Amplify-and-Forward Cooperative Channel,
IEEE Trans. Inform. Theory,vol. 53, no 2, Feb 2007.
[14] S. Yiu, R. Schober, L. Lampe,"Differential Distributed
Space-Time Block Coding", IEEE PacRim conf, 2005.
III - 416
- 1 Tx 6noe
Of
10mUi
2 3 4 5 6
PdB
18 19 20
Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on April 3, 2009 at 13:17 from IEEE Xplore.  Restrictions apply.


A coding scheme for wireless networks with multiple antenna nodes and no channel information

Crossref

A Coding Scheme for Wireless Networks with Multiple Antenna Nodes and No Channel Information

Cooperative Diversity in Wireless Relay Networks with Multiple-Antenna Nodes&quot;,

Differential space-time modulation,&quot;

Distributed Differential Space-Time Coding for Wireless Relay Networks&quot;,

Distributed space-time coding in wireless relay networks,&quot;

Distributed SpaceTime Coded Protocols for Exploiting Cooperative Diversity in Wireless Networks,&quot;

Hassibi,&quot;A Coding Strategy for Wireless Networks with no Channel Information&quot;,

Hassibi,&quot;An Algebraic Coding Scheme for Wireless Relay Networks with Multiple-Antenna Nodes&quot;, subm. to IEEE Trans. Signal Proc.,

Horadam,&quot;A generalised Hadamard transform&quot;,

Lampe,&quot;Differential Distributed Space-Time Block Coding&quot;, IEEE PacRim conf,

Non-Coherent Distributed Space-Time Processing for Multiuser Cooperative Transmissions,&quot;

On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels,&quot;IEEE Trans.

Optimal Space-Time Codes for the Amplify-and-Forward Cooperative Channel,

Partially-coherent distributed space-time codes with differential encoder and decoder,&quot;

Sweldens,&quot;Differential unitary space time modulation&quot;,

http://authors.library.caltech.edu/13767/1/OGGicassp07a.pdf

A coding scheme for wireless networks with multiple antenna nodes and no channel information

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Crossref