Decentralized Power Control with Two-way
Training for Multiple Access
Gajanana G Krishna and Srikrishna Bhashyam
Dept. of Electrical Engineering
Indian Institute of Technology-Madras
Chennai, India
gajananagk@iitm.ac.in, skrishna@ee.iitm.ac.in
Ashutosh Sabharwal
Dept. of Electrical & Computer Engineering
Rice University
Houston, TX 77005
ashu@rice.edu
Abstract— In this work, we analyze the diversity-multiplexing
performance of a MIMO multiple access wireless system with
non-cooperating transmitters. Each of the transmitters and re-
ceiver use noisy and mismatched versions of the channel estimate
to implement decentralized power control. While accounting
for the resources consumed in training, we show that with
relatively simple power control, regardless of the number of
transmitters, we can achieve double the maximum diversity order
of a system with no instantaneous channel state information at
the transmitters. Intuitively, the gain can be attributed to using
temporal degrees of freedom enabled by power control without
coding over multiple coherence intervals.
I. INTRODUCTION
In scenarios where many users attempt to communicate to a
single receiver, multiple antenna systems have been employed
to increase reliability or the supported data rates. A tradeoff be-
tween the two competing objectives when there is no channel
state information at the transmitter (CSIT) was proposed as the
diversity-multiplexing trade-off for the point-to-point MIMO
link [1] and later extended to the multiple access channel
(MAC)[2]. For quasi-static fading channels, it is known that
the order of diversity is the key to link-level performance.
To this end, several forms of feedback have been explored to
provide CSIT and improve the diversity-multiplexing trade-off
(DMT) in both point-to-point MIMO [3],[4] and MAC [5].
However, none of them account for errors and resource con-
sumption on the feedback channel.
A two-way channel formulation was used by the authors
in [6] to study the impact of errors in feedback while fully
accounting for all resource usage in forward and feedback
channels. The two-way formulation is suitable for systems
where the channel is symmetric or SNR-symmetric (same SNR
in both forward and reverse direction) on a per-link basis,
which is often true in time-division duplex systems. For the
case of SIMO channel (single input multiple output), the au-
thors showed that the maximum diversity order can be doubled
compared to a system which employs no feedback. The key
contribution in [6] was the two-way channel formulation and
an analysis which accounts for the mismatch in channel state
information at the transmitter and receiver.
In this paper, we extend the model of [6] to a MAC scenario.
The emphasis in the current work is on understanding in the
multi-user context how mismatch in the knowledge of channel
at the transmitters and receiver affects the system performance,
and if any of the diversity order gains predicted by perfect
feedback [3], [4] are achievable with information mismatch.
With K non-cooperating m-antenna transmitters adopting
decentralized power control and sending messages to a single
antenna at the receiver, we show that we can obtain single
user performance and nearly a two-fold improvement at all
multiplexing gains. The MISO (Multiple Input Single Output)
model is then extended to the general MIMO scenario where
we characterize an achievable DMT for low multiplexing
gains. In particular, we show that in a MIMO MAC system
with m antennas at each of the transmitters and n antennas
at the receiver, we can attain a maximum diversity of 2mn
regardless of the number of transmitters communicating to
the receiver, in contrast to only mn with no CSIT [2]. While
analyzing the multi-user DMT for MIMO, we also furnish
one possible method to extend [6] to the point-to-point MIMO
scenario for low multiplexing gain.
In Section II, we describe the two-way training model
and define outage events for the non-coherent receiver with
training. Section III contain the derivation of DMT for MISO
MAC and and Section IV extends the DMT to MIMO MAC.
We conclude in Section V.
II. S
YSTEM MODEL
A. Channel Model
We consider a multiple access channel with K non-
cooperating transmitters communicating independent mes-
sages to a single receiver. The receiver and each of the
transmitters transmit over the same channel in a time-division
duplexed (TDD) manner. Each transmitter has an array of m
transmit antennas and the receiver has an array of n receive
antennas. We refer to such a system as a K transmitter m × n
MIMO multiple access channel. We assume a slow fading
scenario where the block length of time over which the channel
stays constant equal to l symbols is assumed to be long enough
to make outage errors dominate the total error probability for
the sytem [1][2], l−T
o
≥ Km+n−1 where T
o
is the training
overhead corresponding to the number of symbols used for
training either the receiver or a transmitter.
The channel between transmitter i and the receiver is
represented by the n × m matrix H
it
. Likewise the channel
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
609978-1-4244-2571-6/08/$25.00 ©2008 IEEE
between the receiver and transmitter i is represented by H
T
ir
where notation A
T
is taken to mean the transpose of A.Wesay
that H
it
and H
ir
are symmetric and assume H
it
= H
ir
= H
i
.
In the current work, the proposed protocol requires only SNR-
symmetry such that the singular values of H
it
and H
ir
are
the same. We statistically model {H
i
}
i=1,...,K
to be i.i.d.
and unit variance zero mean circularly symmetric complex
Gaussian (ZMCSCG) entries, the richly scattered Rayleigh
fading environment.
The signal from the K transmitters to the receiver is given
by
Y
r
=
K
i=1
H
i
X
it
+ N
r
, (1)
where X
it
∈ C
m×1
is the complex, information bearing
symbol from the transmitter i, Y
r
∈ C
n×1
is the output vector
and N
r
∈ C
n×1
is the additive noise. In the training phase,
the training signal from the receiver to the i
th
user is given
by
Y
it
= H
T
i
X
r
+ N
it
, (2)
where X
r
∈ C
n×1
is the signal sent by the receiver (e.g.
feedback or training), Y
it
∈ C
m×1
is the corresponding signal
received at transmitter i and N
it
∈ C
m×1
is the additive noise.
The noise at each of the receive and transmit antennas at each
time is i.i.d. unit variance ZMCSCG.
B. Training and Power Control
We build on the model of two-way training in [6] which
allows us to accurately model resource usage and address
mismatched information in a tractable fashion. The training
symbols, which adhere to the same power constraints as data
symbols, are transmitted at a constant power equal to the
average power constraint of the transmitter denoted by
P .
Phase One of the training protocol begins with the trans-
mission of nτ
RT
training symbols (τ
RT
from each of the
receive antennas) from the receiver to the K transmitters.
The transmitters form the channel estimates as
H
i
∀ i =
1, 2,...,K. The error in estimation
H
i
is given by
H
i
=
H
i
+
H
i
. (3)
Then each of the K transmitters independently utilizes its
estimate
H
i
to decide the power control P(
H
i
)=κp(
H
i
)
where κ ∝
P and p(
H
i
) is dependent on the estimate
H
i
. The
receiver does not know the channel estimates {
H
i
}
i=1,2,...,K
at the end of Phase One of training.
In Phase Two, the K transmitters take turns sending mτ
TR
training symbols to the receiver. For reasons explained in
detail in [6], the receiver tries to estimate the power controlled
channels G
i
=
p(
H
i
)H
i
. The power controlled channel
is again split into two parts
G
i
, the estimate of the power
controlled channel, and
G
i
, the estimation error, such that
G
i
=
G
i
+
G
i
. (4)
The entries of
G
i
and
H
i
are ZMCSCG with variance
σ
2
{
G
i
,
H
i
}
=
1
τ
{T R,RT }
P
(5)
meeting the Cram
`
er-Rao bound. Phase Two of training is
followed by (l − nτ
RT
− Kmτ
TR
) symbols of encoded data.
C. Outage Definition and DMT
Making the power control in X
i
explicit, we say that
X
it
=
P
H
i
S
it
where S
it
is the unit power complex
signal prior to power control. Let G =[G
1
G
2
···G
K
],
G =
[
G
1
G
2
···
G
K
] and R
P
=
R
1
(P ) R
2
(P ) ...R
K
(P )
where R
i
(P ) is the data rate from the i
th
user. Also let
α =(l − nτ
RT
− Kmτ
TR
)/l account for rate loss due to
training overhead.
Using the ideas of [7, Section 3,4] for a non-coherent point-
to-point MIMO link, we define outage for the multiple access
non-coherent receiver with training. By an outage event, it is
indicated that the channel is so poor that given an estimate
of the channel state at the receiver, the target data rate is not
supported at least for a subset of the users.
Definition 2.1: For a multiple access channel with K trans-
mitters, each equipped with m transmit antennas and a receiver
with n receive antennas, the outage event is
O
W
O
W
. (6)
The union is taken over all subsets W⊆{1, 2,...,K}, and
O
W
ε
G
: I(X
W
; Y |X
W
c
,
G =
G) <α
i∈W
R
i
,
(7)
where ε
G
is the event that the power controlled channel for
K users G is estimated as
G, X
W
=[X
T
i
1
t
X
T
i
2
t
···X
T
i
|W|
t
]
T
∈
C
|W|m×1
contains the input signals and fading coeffi-
cients respectively corresponding to the users in W =
{i
1
,i
2
,...,i
|W|
}.
In the above definition, the outage event O
W
corresponds
to correct decoding of information from users in W
c
and
loss of information from users in W. When O
W
occurs, the
receiver would have decoded X
W
c
transmitted by users in W
c
correctly. The receiver can without loss of optimality, cancel
its contribution from the received signals. Using (4), we can
write the resultant channel as
Y
W
=Y −
i∈W
c
G
i
S
i
=
i∈W
G
i
S
i
+
K
j=1
G
j
S
j
+ N
r
.
In order to capture the asymptotic performance of our
system, we analyze the diversity-multiplexing tradeoff (DMT)
derived in [1], where diversity, d is
d ≡− lim
P →∞
log( Π(P, R(P )))
log( P )
, (8)
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
610
which describes the rate at which Π, the probability of outage
event O defined in (6) and (7), falls with
P at large P for
a particular target rate, R
P
. It is to be noted that we
fix a common diversity order requirement for all the users.
Multiplexing is defined as
r
i
≡ lim
P →∞
R
i
(P )
log( P )
, (9)
where r
i
quantifies the dependence of the target rate on the
average power constraint. We consider the symmetric case
where all users have the same rate requirement. The extension
to the general case with varying r
i
follows on the lines of [2].
From [2], we also see that
Pr(O)
.
= Pr(O
W
∗
) (10)
where W
∗
is the subset of {1, 2,...,K} with the slowest
decay rate of Pr(O
W
)
1
.
We consider Gaussian codes to determine an achievable
bound for the exponent of outage probability. From [8, Theo-
rem 1], we get
I(X
W
; Y |X
W
c
,
G =
G) = log det
⎛
⎜
⎝
I
n
+
κ
i∈W
G
i
G
†
i
mσ
2
V
⎞
⎟
⎠
(11)
where σ
2
V
=1+
κTr
[
K
j=1
G
j
G
†
j
]
mn
and I
n
is the n dimensional
identity matrix. From (11), and noting that {
G
i
} are inde-
pendent and identically distributed, the outage probability for
event O
W
with |W| = s is given by
Π
m,n
s
= Pr
log det
I
n
+
κ
s
i=1
G
i
G
†
i
mσ
2
V
<srlog
P
.
(12)
For the current work, we assume a joint ML detector. When
all the users have the same diversity order requirement, a joint
ML detector fares just as well as an individual ML detector
from the arguments given in [2, Section VII]. We note the
following Lemma extending [2] to the non-coherent receiver
with training.
Lemma 2.2: The curve d
ˆ
R
K,m,n
(
r
α
) for a non-coherent re-
ceiver with training and no side information at the transmitters
is the DMT derived in [2] shifted by a constant scaling factor
α
=(l − Kmτ
TR
)/l following the arguments of [7].
III. C
HARACTERIZATION FOR MISO MAC
We first consider the case where n =1, a MISO system.
Let all the transmitters have an equal multiplexing gain of r
where
i
R
i
= Kr log(P).
Theorem 3.1: For the multiple access channel considered in
the current work, with a single antenna at the receiver, n =1,
the achievable DMT is given by
d
∗
MAC,m,1
(r)=
2 −
r
α
m 0 ≤ r<
α
K
d
ˆ
R
K,m,n
(
r
α
)
α
K
≤ r ≤
α
K
(13)
1
We use a(x)
.
= b(x) to mean lim
x→∞
a(x)
x
= lim
x→∞
b(x)
x
. Similar
definitions hold for
˙
≤,
˙
≥.
where α =(l − τ
RT
− Kmτ
TR
)/l and α
=(l − Kmτ
TR
)/l
account for the loss of rate due to training and d
ˆ
R
K,m,1
(
r
α
)
refers to the DMT for a MAC system with no CSIT and a
trained receiver.
Proof: When n =1, we adopt the power control
motivated by [6]
P
H
i
= κ
1
γ
i
, (14)
where γ
i
=
H
i
H
†
i
. It can be shown that the power control
algorithm in (14) is stable for m>1. Also σ
2
V
=1+
κ
ζ
m
where
ζ
=
K
j=1
G
j
G
†
j
. We can therefore reduce (12) to
Π
m,1
s
= Pr
log
1+
κ
s
i=1
ζ
i
m + κ
ζ
<
sr
α
log
P
, (15)
where
ζ
i
=
G
i
G
†
i
. Rearranging (15), we get
Π
m,1
s
= Pr
κ
s
i=1
ζ
i
<
P
sr
α
− 1
m + κ
ζ
. (16)
Putting s =1, we obtain the exponent of the outage probability
for a single user given in [9]. For s ≥ 1, we perform the
following manipulations.
We bound (16) by the sum of two terms that intuitively
separate the effects of uncertainty in the channel state at the
transmitters and the receiver. Let δ be a positive constant.
Π
m,1
s
˙
≤Pr
κ
s
i=1
ζ
i
<P
(
sr
α
+δ
)
κ
ζ
<mP
δ
+ Pr
κ
s
i=1
ζ
i
<P
sr
α
2κ
ζ
κ
ζ
≥ mP
δ
˙
≤ Pr
κ
s
i=1
ζ
i
<P
(
sr
α
+δ
)
Π
m,1
s1
+ Pr
κ
ζ
>mP
δ
Π
m,1
s2
. (17)
The first term Π
m,1
s1
arises out of uncertainty in the transmit-
ters’ Channel State Information (CSI) while the second term
Π
m,1
s2
comes about from the uncertainty of CSI at the receiver.
We notice a slight similarity between (41) in [6] and (17).
However, as we shall see, unlike in [6], the partitioning in (17)
allows us tackle the multi-user situation better by removing the
effect of estimation error at the receiver completely.
Turning our attention to the second term Π
m,1
s2
, we see that
Pr
κ
ζ
>mP
δ
=
∞
P
δ
κ
f
ζ
(x)dx
(a)
=
∞
P
δ
κ
1
Γ(m)σ
2
G
x
σ
2
G
!
Km−1
e
−
x
σ
2
G
dx
(b)
.
=
P
δ
Km−1
e
−P
δ
. (18)
In (a) we have substituted the distribution for the χ
2
-square
variable with 2Km degrees of freedom and in (b), we have
substituted (5). Plugging the exponentially decaying error
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
611
probability of (18) into (8), we see that any δ>0 will
yield infinite diversity. The physical implication is that the
estimation error at the receiver will not alter the DMT apart
from imposing a training overhead. The error in channel state
information at the transmitter alone is the limiting factor.
We note that substituting for G in (4), we have
G
i
=
1
"
γ
i
H
i
−
G
i
=
1
"
γ
i
H
i
, (19)
where
H
i
is the receiver’s normalized estimate of the channel
between transmitter i the receiver. Equivalently, using (3), we
can write
H
i
in terms of
H
i
, the estimate of the channel at
the transmitter
H
i
= H
i
−
"
γ
i
G
i
=
H
i
−
H
i
, (20)
where
H
i
=
"
γ
i
G
i
−
H
i
is another noise matrix with
ZMCSCG entries and variance
σ
2
H
i
=
1
τ
RT
P
+
γ
i
τ
TR
P
.
=
P
−1
. (21)
In essence, the estimate of the channel state at each transmitter
is a noisy version of the normalized estimate of the channel
state at the receiver with the mismatch given by σ
2
H
i
.
Let γ
i
=
#
H
i
#
H
†
i
. From the definition of (19), we write
ζ
i
=
γ
i
γ
i
. Noting that κ ∝ P , the first term, Π
m,1
1s
, in (17) can
be recast as
Π
m,1
1s
.
=Pr
s
i=1
P
(
1−
sr
α
−δ
)
γ
i
γ
i
< 1
(22)
(c)
≤
s
$
i=1
Pr
P
(
1−
sr
α
−δ
)
γ
i
γ
i
< 1
!
(d)
.
=
⎧
⎨
⎩
P
m
(
2−
sr
α
−δ
)
!
s
0 ≤ r<
α(1−δ)
s
P
0
r ≥
α(1−δ)
s
where (c) arises out of the independence of {
γ
i
γ
i
}
i=1,...,s
and
(d) comes about from the analysis in [9] using (20) and (21).
Since the analysis of an upper bound to the outage proba-
bility is valid for any δ>0, we make δ → 0
+
. Recalling that
Π
m,1
s
˙
≤Π
m,1
1s
,weget
Π
m,1
s
˙
≤
P
ms
(
2−
sr
α
)
0 ≤ r<
α
s
P
0
r ≥
α
s
(23)
We see from (23) that the DMT curve for error events O
W
with |W| > 1 lie above the single user DMT curve up to
r ≤
α
s
. Therefore, from (10), the resultant DMT is as given
in (13).
Remark 3.2: Unlike the case of no CSIT [2], where the
DMT is divided into lightly loaded and heavily loaded regions
with single user and K user performances respectively, we
get single user performance for almost the full range of
permissible multiplexing gain
0 0.05 0.1 0.15 0.2 0.25
0
0.5
1
1.5
2
2.5
3
3.5
4
Multiplexing Gain
Diversity
r=α/K
r=α’/K
r=α’/(K+1)
With noisy
CSIT
No CSIT
Fig. 1. Achievable DMT of a 4 transmitter 2 × 1 MAC system with τ
TR
=
τ
RT
=10symbols and coherence length l = 1000
We have plotted the DMT for a 4 transmitter 2 × 1 multiple
access channel in Figure 1 and compared it with the case
where there is no side-information at the transmitters.
IV. MIMO MAC
FOR LOW MULTIPLEXING GAINS
We now consider a general MIMO system with n ≥ 1
and show that at low multiplexing gains, there is a substantial
improvement in the DMT for the general scenario as well.
Theorem 4.1: For the multiple access channel considered in
the current work, with a general MIMO system, the achievable
DMT is given by
d
∗
MAC,m,n
(r)=
2 −
r
α
mn 0 ≤ r<
α
K
d
ˆ
R
K,m,n
(
r
α
)
α
K
≤ r ≤ min{m,
n
K
}α
(24)
where α =(l −nτ
RT
− Kmτ
TR
)/l and α
=(l − Kmτ
TR
)/l
account for the loss of rate due to training and d
ˆ
R
K,m,n
(
r
α
)
refers to the DMT for a MAC system with no CSIT and a
trained receiver.
Proof: Building on the analysis for a MISO MAC
system in Section III, we now construct an upper bound
for Π
m,n
s
for s =1, 2,...,K. At transmitter i, let
H
i
=
[
H
(1)T
i
H
(2)T
i
···
H
(n)T
i
]
T
, where {
H
(j)
i
}
n
j=1
represent the es-
timates of the n parallel MISO channels from transmitter i to
the receiver. Each transmitter implements a worst case power
control
P
H
i
= κ
1
min
j
γ
j
i
, (25)
where γ
j
i
=
H
(j)
i
H
(j)†
i
. Note that κ ∝ P although different
from (14). Let the receiver implement selection and decode the
messages at the n antennas separately. As we are operating at
low multiplexing gains (Kr < α), we do not use the additional
degrees of freedom provided by the n antennas and instead
use them to improve reliability (diversity). Clearly, the outage
event O
W
for such a scheme occurs when all the n MISO
links are in outage. After removing the effect of receiver
estimation error at each of the the n MISO multiple access
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
612
parallel channels, in the manner of (22), we get
Π
m,n
s
=
n
$
l=1
Pr
s
i=1
P
(
1−
sr
α
−δ
)
γ
l
i
min
j
γ
j
i
< 1
˙
≤
n
$
l=1
Pr
s
i=1
P
(
1−
sr
α
−δ
)
γ
l
i
γ
l
i
< 1
(e)
˙
≤
Π
m,1
s
n
.
=
P
smn
(
2−
sr
α
−δ
)
0 ≤ r<
(1−δ)α
s
P
0
r ≥
(1−δ)α
s
(26)
where in (e) we note that the factors in the product are iden-
tically distributed multiple access MISO for l =1, 2,...,n.
Setting δ → 0
+
, we again see that the DMT curve for error
events O
W
with s = |W| > 1 lie above the single user DMT
curve up to r ≤
α
s
. Therefore, from (10), the resultant DMT
is as given in (24). We switch to a CSI
ˆ
R system for r ≥
α
K
.
In deriving the achievable DMT for multi-user MIMO
channel, we have also shown a method of extending the results
of [6] for a SIMO channel to the single user point-to-point
MIMO channel.
Corollary 4.2: The point-to-point MIMO link with the two-
way training model will achieve the DMT d
∗
MIMO,m,n
(r) given
below.
d
∗
MIMO,m,n
(r)=
mn
2 −
r
α
0 ≤ r<α
d
ˆ
R
MIMO,m,n
(
r
α
) r ≥ α
(27)
where d
ˆ
R
MIMO,m,n
(
r
α
) is the fundamental DMT for a point-to-
point link for a trained receiver [7].
Proof: Put s =1in (26) to get the result.
In Figure 2, we have plotted the DMT for a 4 transmitter
2 × 2 MIMO multiple access channel and compared it with
the case where there is no side-information at the transmitters.
The benefit of single user performance at low multiplexing
gains enables us to approximately double the diversity order
in the range 0 ≤ r ≤
α
K
. Moreover, for any K transmitter
m × n system, at zero multiplexing gain for all transmitters,
i.e. for a fixed rates not varied with power, the diversity order
is 2mn compared with mn in a CSI
ˆ
R system irrespective of
the number of users. Such a gain can be ascribed to using
temporal degrees of freedom obtained from power control
without having to code over multiple uncorrelated coherence
intervals.
Since a multiple access channel cannot outperform a single-
user channel, we see that decentralized power control performs
quite well in our model at low multiplexing gains. For higher
multiplexing gains, centralized power control seems to become
necessary as the bound quickly falls off. Inversion with smaller
eigenvalues, second largest, third largest and so on, might
extend the range of multiplexing gain where our scheme of
decentralized power control is useful. But the analysis is much
more difficult.
Sub-optimal receivers like MMSE or successive cancelation
require the number of receive antennas to be larger than the
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
1
2
3
4
5
6
7
8
Multiplexing Gain
Diversity
r=α/K=0.225
r=min{m,n/K+1}α’
r=min{m,n/K}α’
With noisy CSIT
No CSIT
Fig. 2. Achievable DMT of a 4 transmitter 2 × 2 system with τ
TR
=
τ
RT
=10symbols and coherence length l = 1000
number of transmitters, n>K. However, as noted in [2],
the constraint n>Kis not fundamental. We have shown
that even with a few antennas at the receiver, we can obtain
substantial diversity for an arbitrary number of users over all
total multiplexing gains up to unity,
1
α
Kr < 1, with an optimal
joint ML receiver.
V. C
ONCLUSION
In this work, we analyzed multiple access systems using the
two-way formulation for feedback channel introduced in [6].
Our results clearly indicate that decentralized power control
by non-cooperating transmitters is extremely beneficial at low
multiplexing gains even when training the transmitters con-
sumes extra resources and has errors. The maximum diversity
in our model with noisy and mismatched channel estimates at
the transmitters and the receiver is double that achieved with
no side information at the transmitter [2].
R
EFERENCES
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