Space Diversity for Multi-antenna Multi-relay Channels

Abstract

In this paper we analyze the performance of multiple relay channels when multiple antennas are deployed only at relays. We apply two antenna diversity techniques at relays, namely maximum ratio combining (MRC) on receive and transmit beamforming (TB). We show that with K relays the network can be decomposed into K diversity channels each with a different channel gain, and that the signals can be effectively combined at the destination. We assume that the total number of antennas at all relays is fixed at N. If the total transmit power for all relays are the same as for the source and equally distributed among all the relays, the network capacity will be lower bounded by that of N relay channels each with single antenna, and upper bounded by that of single relay channels with N antennas.

Full text

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Space Diversity for Multi-antenna Multi-relay
Channels
Yijia Fan and John Thompson
Institute for Digital Communications
School of Engineering and Electronics
University of Edinburgh
Edinburgh, EH9 3JL, UK
Email: Y.fan@ed.ac.uk
Abdulkareem Adinoyi
and Halim Yanikomeroglu
Broadband Communications
and Wireless Systems (BCWS) centre
Department of Systems and Computer Engineering
Carleton University
Ottawa, K1S 5B6, Canada
Email: adinoyi@sce.carleton.ca
Abstract—In this paper we analyze the performance of
multiple relay channels when multiple antennas are deployed
only at relays. We apply two antenna diversity techniques at
relays, namely maximum ratio combining (MRC) on receive and
transmit beamforming (TB). We show that with K relays the
network can be decomposed into K diversity channels each with
a different channel gain, and that the signals can be effectively
combined at the destination. We assume that the total number of
antennas at all relays is fixed at N. If the total transmit power for
all relays are the same as for the source and equally distributed
among all the relays, the network capacity will be lower bounded
by that of N relay channels each with single antenna, and upper
bounded by that of single relay channels with N antennas.
I. INTRODUCTION
It is widely believed that ad hoc networking [1] or multi-
hop cellular networks [2] are important new concepts for
future generation wireless systems [3], where either mobile
or fixed nodes (often referred to as relays) are used to help
forward the information to the desired user. One advantage
of these structures are that it is possible to unite multiple
relays in the network as a “virtual antenna array” to forward
the information cooperatively, while appropriate combining at
the destination realizes diversity gain. The diversity achieved
in this way is often named as user cooperation diversity
or cooperative diversity [4], as it mimics the performance
advantages of multiple-input multiple-output (MIMO) systems
[5] in exploiting the spatial diversity of the relay channels. The
performance limits of space-time codes, which can exploit
cooperative diversity, are discussed in [6]–[8] for single-
antenna relay networks. For multiple-antenna relay channels
where every terminal in the network can be deployed with
multiple antennas, studies are mainly concentrated on spatial
multiplexing systems [9]–[11].
In this paper we exploit the spatial diversity of the relay
channels in a different way from space-time codes based
approach. We apply two kinds of antenna combining tech-
niques at the relay, namely maximum ratio combining (MRC)
[12] for reception and transmit beamforming (TB) [13] for
transmission. Those techniques were often used in point-to-
point wireless links to enhance signal-to-noise ratio (SNR)
at the output of the receiver by deploying multiple antennas
at either transmitter or receiver. In a relay context, we move
the multiple antennas to the relays, while the source and
the destination are only deployed with a single antenna. Our
investigation is based on digital relaying, where the relays
decode, re-encode and re-transmit the signals. We show that
the network with K relays can be decomposed into K diversity
channels each with different channel gain, and the signals from
all K branch can be effectively combined at the destination.
We derive the capacity bounds for this signal combining
techniques. Our analysis results can be applied to both ergodic
capacity and outage capacity [14] performance.
The rest of this paper is organized as follows. In Section
II, the basic system model and assumptions are introduced.
Section III introduces the signal combining techniques. The
capacity performance analysis are made in section IV. Sec-
tion V presents and discusses simulation results and finally,
conclusions are drawn in Section VI.
II. SYSTEM MODEL
We consider a two hop network model with one source, one
destination and K relays. We ignore the direct link between
source and destination. We also assume that total transmit
power of the source and relays are the same and that it is
equally distributed among the relays. Each relay processes the
received signals independently. We assume that the source and
destination are deployed with single antennas, while relay k is
deployed with mkantennas. We assume that the total number
of antennas at all relays is fixed to N. This can be expressed
as
K
?
We restrict our discussion to the case where the channels are
slow, frequency-flat fading. The data transmission is over two
times slots using two hops. In the first transmission time slot,
the source broadcasts the signal to all the relay terminals. The
input/output relation for the source to the kth relay is given
by
rk=√ηhks + nk,
k=1
mk= N.
(1)
(2)

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where rk is mk× 1 receive signal vector. η denotes the
transmit power at the source. The s is the transmit signal
with covariance 1 and nk is the mk× 1 complex circular
additive white Gaussian noise vector at relay k with identity
covariance matrix Imk. The vector hkis the mk× 1 channel
transfer matrix from source to the kth relay and can be further
expressed as
hk=√αk˜hk,
(3)
where each entry of˜hkare identically independent distributed
(i.i.d) complex Gaussian random variables with unit variance.
Each factor αk contains the pathloss and can be written as
αk= x−γ
relay k. The scalar γ denotes the path loss exponent. In the
second hop, each relay processes its received signals and re-
transmits them to the destination. The signal received at the
destination can be written as:
k, where xkis the distance between the source and
y =
K
?
i=1
gkdk+ nd,
(4)
where the vector gk is the channel matrix from kth relay to
the destination, which might also be written as:
?
where each entry of ˜ gkis an i.i.d. complex Gaussian random
variables with unit variance. The scalar βk also contains the
pathloss from the kth relay to the destination. The scalar ndis
the complex additive white Gaussian noise at the destination
with unit variance. The vector dk is the transmit signal
vector at relay k, which should meet the total transmit power
constraint:
E?dk?2
where ?•?Fdenotes the Frobenius norm. We assume a co-
herent relay channel configuration context where the kth relay
can obtain full knowledge of both backward channel vector
hk and forward channel vector gk. For fair comparison, we
also assume that for each channel realization, either backward
or forward channel coefficients for all N antennas remains
the same regardless of the number of relays K. It will not be
difficult to see that the conclusions on either Ergodic capacity
or outage capacity in this paper also hold if we extend the
discuss to a more general case where each antenna is fixed in
the network. Fig. 1 gives a description for the system model.
gk=
βk˜ gk,
(5)
?
F
?
≤ηmk
N
,
(6)
III. ANTENNA DIVERSITY TECHNIQUES IN RELAY
CHANNELS
In this section we apply MRC and TB techniques to the
system model described in section II. We assume each relay
performs MRC of the received signals, by multiplying the
received signal vectors by the vector hH
at output of the relay receiver is given by
k
??hk?F. The signals
?
?
˜ rk=
?
?
?
?η
mk
?
i=1
|hi,k|2s +
mk
i=1h∗
i,kni,k
mk
?
i=1|hi,k|2
(7)
Source
Relay 1
Relay 2
Relay K
1
2
3
N
Destination
Fig. 1.
each deployed with 1 antenna. Totally N antennas are deployed at K relays.
For each channel realization, either backward or forward channel coefficients
for all N antennas remains the same regardless of the number of relays K.
System model for a two hop network: Source and destination are
where hi,kdenotes the ith antenna at relay k, and ni,kdenotes
the noise factor for ith receiver input branch. The SNR at the
output of the receiver can be written as:
ρmk
k
= η
mk
?
i=1
|hi,k|2.
(8)
After the relays decode the signals, each relay then performs
TB of the transmitted signals. If we denote the transmitted
signals as dkwith unit variance, the transmitted signal vector
dkfor relay k can be written as
?ηmk
The destination receiver simply detects the combined signals
from all K relays. If we adjust the transmission data rate so
that the signals are correctly decoded at all the relays (i.e.
dk= s), the output signal at the destination can be written as:
?
N
i=1
dk=
N
gH
k
?gk?F
dk.
(9)
y = s
K
?
k=1
?
?
?ηmk
mk
?
|gi,k|2+ nd= s
K
?
k=1
˜ gk+ nd (10)
It can be seen from (10) that by applying antenna diversity
schemes at relays, the networks can be decomposed to K
diversity channels each with channel gain ˜ gk. The output SNR
at the destination receiver can therefore be written as:


When all the relays are deployed with a single antenna, there is
no traditional maximum ratio combining gain at the relays and
the destination. However, the destination still observes a set of
equal gain combined [15] amplitude signals from all relays.1
ρmk
d
=
K
?
k=1
?
?
?
?ηmk
N
mk
?
i=1
|gi,k|2


2
.
(11)
1Different from [15], the equal gain combining for relay channels is applied
at the transmitter instead of the receiver.

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Since we assume that the backward and forward channel
coefficients for each antennas are kept same for different
number of K and mi. The output SNR at the destination can
be rewritten as
?K
k=1
when all the antenna are deployed in one relay (i.e. K = 1
and m1= N), full diversity gain is achieved among all the
N antennas at the relay and also at the destination. The SNR
can be rewritten as
ρ1
d=η
N
?
mi
?
i=1
|gi,k|
?2
;
(12)
ρN
d= η
K
?
k=1
mi
?
i=1
|gi,k|2
(13)
IV. CAPACITY PERFORMANCE
In this section we derive the capacity bounds for the scheme
proposed in previous section. The network capacity for digital
relaying can be written as
Cmk
D
= min?C1,m1
Ck,mk
r
r
,C2,m2
r
,···,CK,mk
k)
r
,Cmk
d
?
(14)
where
Shannon capacity from source to relay k channel, and
Cmk
d
= 0.5log2(1 + ρmk
from relays to destination channels. The factor 0.5 denotes
the half bandwidth compared with non-relay channels.
We firstly analyze channel capacity from the relays to
destination link by bounding the ρmk
the destination.
Lemma 1: For any mk, ρ1
Proof: See Appendix.
From Lemma 1, we can see that
= 0.5log2(1 + ρmk
denotingthe
d) denoting the Shannon capacity
d,i.e. the output SNR at
d≤ ρmk
d
≤ ρN
d.
C1
d≤ Cmk
d
≤ CN
d,
(15)
where C1
nels when K = N, and CN
to destination channels when K = 1. Now also considering
the capacity from the source to relays link and extending the
analysis to the whole network scenario, we have the following
theorem:
Theorem 1: If we denote the network capacity for K = N
as C1
Proof: Considering the SNR ρmk
kth relay link, if we denote it as ρ1
K = 1, it can be shown that
min?ρ1
Therefore, we have the following:
min?C1,1
Combining (17) and (15), we thus complete the proof.
From the above analysis we have shown that for the signal
combining techniques discussed in the paper, the network
capacity will be lower bounded by that of N relay channels
each with single antenna, and upper bounded by that of a
ddenotes the capacity for relays to destination chan-
ddenotes the capacity for relay
Dand for K = 1 as CN
D, for any mk, C1
D≤ Cmk
for the source to the
nfor K = N and ρN
D
≤ CN
D.
k
1for
n
?≤ min(ρmk
?≤ min?C1,m1
k) ≤ ρN
1.
(16)
r ,···,CN,1
rr
,···,CK,mk
r
?≤ C1,N
r
.
(17)
−10 −8−6 −4−20
P
246810
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
10% outage capacity
K=6,m=1
K=3,m=2
K=2,m=3
K=1,m=6
(a) 10% outage capacity
−10−8−6−4−20
P
246810
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Ergodic capacity
K=6,m=1
K=3,m=2
K=2,m=3
K=1,m=6
(b) Ergodic capacity
Fig. 2.
relays K, while each relay is deployed with m antennas. (a) 10% outage
capacity. (b) Ergodic capacity.
Capacity of single MIMO relay channels for different number of
single relay channel with N antennas. This means that even
there are more relays, the increased “equal gain combining”
gain at the destination can not compensate for the loss of
maximum ratio combining gain at the relay and the destination
when numbers of antennas at each relay are reduced.
V. SIMULATION RESULTS
We calculate the both ergodic capacity and 10% outage
capacity (in bits per channel use) for 1000 channel realizations
regarding different values of η, denoted as P in the figures. In
this simulation example we assume that the distance between
source and destination is normalized. The relays are uniformly
and randomly located in the middle region between the source
and relays. Therefore xk is set to 0.5. We assume the total
number of antennas at relays (N) is 6 and we also assume
that all K relays have the same number of antennas m.
Fig. 2 shows the capacity performance. We can see that for
different (K,m), the capacity is always upper bounded by
(1,6) and lower bounded by (6,1). These results verify the
analysis made in this paper. Furthermore, we can see through
the simulation that larger m and small K might give larger

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benefit, since larger m allows more freedom of cooperation
among the antennas at each relay. Therefore when m reaches
N (K reduces to 1), full cooperation are made among all the
antennas to give rise to the best performance.
VI. CONCLUSIONS
In this paper we analyze the performance of multiple
relay channels when multiple antennas are deployed only
at relays. We apply antenna diversity techniques at relays
which are known as maximum ratio combining and transmit
beamforming. We show that with K relays the network can be
decomposed into K diversity channels each with a different
channel gain, while the signals can be effectively combined at
the destination. If we assume that the total number of antennas
at all relays are fixed to N and total transmit power at all relays
are normalized, the network capacity will be lower bounded by
that of N relay channels each with single antenna, and upper
bounded by that of single relay channels with N antennas.
APPENDIX
PROOF OF Lemma 1
We firstly prove that ρ1
d≤ ρmk
d. We write the following
?
ρmk
d
−
?
ρ1
d=
K
?
k=1







?
?
?
?
?ηmk
N
mk
?
??
i=1
|gi,k|2
?
Ak
−
?η
?
N
mk
?
??
i=1
|gi,k|
?
Bk







.
(18)
To compare Akwith Bk, we write
A2
k− B2
k=η
N

mk
?
mk
?
i=1
|gi,k|2−
?mk
i=1
?
?
|gi,k|
?2

(19)
=η
N
(mk− 1)


mk
?
i=1
|gi,k|2
−η
N
mk
?
i,j=1;i?=j
|gi,k||gj,k|

.
(20)
Note that
(mk− 1)
mk
?
i=1
|gi,k|2=
mk
?
i=1
mk
?
mk
?
j=1,j?=i
|gj,k|2
(21)
=0.5
i,j=1;i?=j
?
|gi,k|2+ |gj,k|2?
.(22)
So (20) can be further written as:
A2
k− B2
k=η
2N
mk
?
mk
?
i,j=1;i?=j
?
|gi,k|2− 2|gi,k||gj,k| + |gj,k|2?
=η
2N
i,j=1;i?=j
(|gi,k| − |gj,k|)2≥ 0.
So Ak≥ Bkand therefore ρ1
d≤ ρmk
d.
Next we prove that ρN
d≥ ρmi
d. For simplicity, we denote
ak=
mk
?
i=1
|gi,k|2
(23)
in equation (11) and (13). Then ρN
d− ρmi
d
can be written as
ρN
d−ρmi
d
=η
N


K
?
k=1
(N − mk)ak−
K
?
i,j=1;i?=j
√mimjaiaj

.
(24)
Note the constraint by (1) in section II, we have the following:
(N − mk) =
K
?
i=1,i?=k
mi.
(25)
Putting (25) into (24), we have the following:


Note the following:
ρN
d−ρmi
d
=η
N
K
?
k=1
K
?
i=1,i?=k
miak−
K
?
i,j=1;i?=j
√miaimjaj

.
(26)
K
?
k=1
K
?
i=1,i?=k
miak=
K
?


i,j=1;i?=j
(miaj)
=0.5
K
?
i,j=1;i?=j
(miaj) +
K
?
i,j=1;i?=j
(mjai)

.
We can further write (26) as follows:
ρN
d− ρmi
d=η
2N
K
?
?
i,j=1;i?=j
(miaj)
−η
N
K
i,j=1;i?=j
√miajmjai
+
η
2N
K
?
i,j=1;i?=j
(mjai)
=η
2N
K
?
d≤ ρmk
REFERENCES
i,j=1;i?=j
?√miaj−√mjai
≤ ρN
?2.
Therefore ρN
d≥ ρmi
d
and ρ1
d
d.
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