Outage diversity of MIMO block-fading channels with causal channel state information.

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Outage Diversity of MIMO Block-Fading Channels
with Causal Channel State Information
Khoa D. Nguyen
Institute for Telecommunications Research
University of South Australia
Mawson Lakes, SA 5095
{khoa.nguyen,nick.letzepis}@unisa.edu.au
Nick LetzepisAlbert Guill´ en i F` abregas
Department of Engineering
University of Cambridge
Cambridge CB2 1PZ UK
guillen@ieee.org
Lars K. Rasmussen
Communication Theory Lab
Royal Institute of Technology
Stockholm, Sweden
lars.rasmussen@ee.kth.se
Abstract—We study the outage diversity of the multiple-
input multiple-output (MIMO) Rayleigh block-fading channel
when causal channel state information (CSI) is available at the
transmitter (CSIT). Within this setting, we consider the optimal
power allocation for blocks b = 1,...,B given perfect CSIT for
blocks 1,...,b−u only, subject to a long-term power constraint.
The parameter 0 ≤ u ≤ B is a fixed arbitrary integer that
determines the delay in acquiring perfect knowledge of the CSI
at the transmitter. Without explicitly solving the optimal power
allocation problem, we derive the outage diversity of the system.
For general 0 ≤ u ≤ B, we derive a simple recursive expression
for computing the outage diversity. For the special case u = 0,
it is shown that the outage diversity is infinite, coinciding with
previously known results. For 1 ≤ u ≤ B, the outage diversity
becomes finite and for the special case of u = 1,2 it can be
expressed in simple closed form.
I. INTRODUCTION
The mitigation of fading is a particularly challenging aspect
in the design of reliable and efficient wireless communication
systems [1]. Methods that attempt to deal with this impairment
depend on many factors, e.g., the behaviour of the fading (co-
herence time/frequency) and system constraints (delay/power).
For systems with no delay constraints or fast fading channels,
the channel can be considered ergodic. In this case, long-
interleaved fixed-rate codes not exceeding the channel capacity
can be employed to ensure an arbitrarily low probability of
error [2,3]. On the other hand, for slowly fading channels with
delay constraints, the codeword may only experience a small
finite number of independent fading realisations and hence the
channel is non-ergodic.
The block-fading channel [2,4] is a simple model that
captures the essence of non-ergodic channels. Here, each
codeword comprises a finite number of blocks, where each
block experiences an independent fading realisation, which
remains constant within a given block. In this case, the
instantaneous input-output mutual information is a random
variable dependent on the underlying fading distribution. For
most fading statistics, the channel capacity is zero in the strict
Shannon sense as there is a non-zero outage probability that a
fixed information rate cannot be supported [2,4]. The outage
This work has been supported by the Sir Ross and Sir Keith Smith
Fund, Cisco Systems; the Australian Research Council under ARC grants
RN0459498, DP0881160; the Royal Society International Travel Grant
2009/R2; and the Swedish Research Council under VR grant 621-2009-4666.
probability is the lowest achievable word error probability
of codes with sufficiently long block length [5]. As such, a
rate-reliability tradeoff exists, whereby for a fixed number of
blocks, a high rate is penalised by a large error probability.
Most works that study the block-fading channel focus on
adaptive transmission techniques in which the power and/or
rate is adapted to the channel conditions subject to system
constraints (see [6] for a recent review). Adaptation, however,
requires a certain degree of knowledge of the channel fades,
also referred to as channel state information (CSI), at the
transmitter and receiver. A large body of works consider
perfect a-causal CSI at the transmitter (CSIT), i.e., the trans-
mitter knows exact values of the fades on all blocks, which
enables adaptation using full CSIT. This approach has practical
relevance for systems exhibiting a set of instantaneous parallel
channels, such as Orthogonal FrequencyDivision Multiplexing
(OFDM) systems. However, the assumption is unrealistic for
slowly time-varying channels where there is a delay in acquir-
ing the CSIT, e.g., free-space optical channels [7]. For these
types of systems it is only realistic to assume that only causal
CSIT, i.e., past channel fades, are available at the transmitter.
Several works have analysed the block-fading channel with
causal CSIT [8–10], where power adaptation algorithms based
on dynamic programming are proposed. However, in [8–10] it
is assumed that perfect CSIT is available up to and including
the current block to be transmitted. In practical systems there
can be additional latency in the transmitter acquiring the CSIT,
e.g. propagation and processing delays. Such additional delays
are our primary motivation in this paper. In particular, we con-
sider the multiple-input multiple-output (MIMO) block-fading
channel with discrete-input constellations, where there is an
arbitrary fixed delay in the availability of perfect CSI at the
transmitter. The transmitter employs optimal power allocation
that minimises the outage probability subject to a long-term
power constraint. The optimal power allocation rule can be
obtained as an extension of the dynamic programs in [8–10],
however, the algorithms may grow prohibitively complex with
increasing delay in acquiring CSIT. Dynamic programming
therefore provides little insight in the outage performance
of systems with causal CSIT. Using recent results from [6,
11], we derive the outage diversity without explicitly solving
the optimal power allocation problem. We show that the
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outage diversity requires the solution to a linear homogeneous
recurrence relation (difference equation) [12], which admits
no closed form in general, but can be easily computed numer-
ically. For the special case of a single block delay, the outage
diversity can be solved in closed form and coincides with
recent results on incremental redundancy, automatic repeat-
request (INR-ARQ) systems [13,14]. For a two-block delay,
the difference equation can be solved analytically, yielding a
closed-form expression for the outage diversity. Furthermore,
the outage diversity of the single-input single-output (SISO)
case can be simply characterised from the Fibonacci series.
In general, our results completely characterise the loss in
outage diversity for an arbitrary fixed delay in obtaining CSIT.
Furthermore, it is also shown that only CSIT with sufficiently
small delay is useful in terms of outage diversity.
II. SYSTEM MODEL
In this paper, we consider a discrete-input MIMO block-
fading channel, with Ntand Nrtransmit and receive antennae,
respectively. Binary data is encoded with a code of rate R bits
per channel use, constructed over an alphabet X ⊆ C. The
resulting transmitted codeword consists of B blocks, where
each block comprises of L vector channel uses of size Nt×1.
We denote xb[l] ∈ XNt×1as the lth transmitted symbol vector
of block b, for l = 1,...,L and b = 1,...,B. The symbols
are assumed to be drawn from X with unit average energy,
i.e. E
?
transpose and IN the N × N identity matrix.
We denote by Hbthe Nr×Ntcomplex channel matrix for
block b = 1,...,B. These matrices are drawn independently
for each block, and remain fixed for the corresponding L
channel uses. In addition, we assume the elements of the
channel matrix are i.i.d. Gaussian random variables (Rayleigh
fading model [1]). At the transmission of block b, we as-
sume the transmitter only has knowledge of H(b−u)
(H1,...,Hb−u), where 0 ≤ u ≤ B is an arbitrary fixed
integer (causal-CSIT). The parameter u models the delay in
the transmitter obtaining the CSIT, due to e.g., propagation
and processing delays.
We assume the receiver has perfect knowledge of H(b)at
transmission block b, and the signal at each receive antenna
is corrupted by independent, zero-mean unit-variance additive
white Gaussian noise (AWGN). Hence, under these assump-
tions, the bth block of Nr× L received noisy symbols is
Yb= HbP
b
Xb+ Wb,
xb[l]x†
b[l]
?
= INt, where (·)†denotes the Hermitian
=
1
2
?
H(b−u)?
b = 1,...,B, (1)
where: Xb ∈ XNt×L; Hb ∈ CNr×Nt; Yb ∈ CNr×L;
Wb ∈ CNr×Lwhose elements are i.i.d zero-mean unit-
variance white Gaussian noise; and Pb
is a diagonal matrix whose ith diagonal element denotes the
power allocated to transmit antenna i of block b. The power
allocation is subject to the long-term power constraint,
?
H(b−u)?
∈ RNt×Nt
+
E
?
1
B
B
?
b=1
tr
?
Pb
?
H(b−u)???
≤ P.
(2)
III. PRELIMINARIES
The channel described by (1) under the quasi-static assump-
tion is not information stable [15] and therefore, the capacity
in the strict Shannon sense is zero. We therefore study the
information outage probability,
Pout(P,R) = Pr
?
1
B
B
?
b=1
IX
?
HbP
1
2
b
?
H(b−u)??
< R
?
,
(3)
which lower bounds the codeword error probability of any
coding scheme [2,4]. In (3), IX(S) denotes the input-output
mutual information of a MIMO block-fading channel with
input constellation X and channel matrix S. With the optimal
Gaussian input constellation,
IX(S) = log2det(INr+ SS†);
(4)
while with a uniform discrete constellation X of size 2M,
IX(S) = MNt−
1
2MNt
x∈XNt
x?∈XNt
?
Ez
⎡
⎣log2
⎛
⎝
?
e−?S(x−x?)+z?2+?z?2
⎞
⎠
⎤
⎦
(5)
Given perfect knowledge of H(b−u), Pb(H(b−u)) is the
solution to the minimisation problem
?minimise
subject to:
Pout(P,R)
?
E
1
B
?B
b=1tr
?
Pb
?
H(b−u)???
≤ P.
(6)
For u = 0, (6) can be solved via dynamic programming [8,
10]. The extension to u > 0 is also possible, although the
problem becomes exceedingly difficult as u increases. As we
shall see, it is possible to examine the asymptotic behaviour
of Pout(P,R) without explicitly solving (6). Particularly, we
study the outage diversity d(R),
d(R) ? lim
P→∞
−logPout(P,R)
logP
.
(7)
For systems with uniform power allocation, the outage di-
versity is duni(R) = BNtNr, achieved with Gaussian input
constellation. With a discrete constellation X of size 2M, the
outage diversity is given by the Singleton bound [6,16]
duni(R) = dS(R) = Nr
?
1 +
?
B
?
Nt−R
M
???
,
(8)
with ?x? being the largest integer not greater than x. Note that
duni(R) is also the outage diversity of systems with short-term
power constraint?B
IV. MAIN RESULTS
For systems with causal CSIT, the outage diversity in both
Gaussian and discrete input cases is a function of duni(R), as
given in the following Theorem.
Theorem 1: Consider transmission at rate R over the
MIMO block-fading channel in (1) with Rayleigh fading. With
b=1tr
?
Pb
?
H((b−u))??
≤ BP.
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long-term power constraint in (2) and CSIT delay 0 < u ≤ B,
the optimal outage diversity is
d(R) = NtNr
ˆb
?
?
b=1
ab+
?
duni(R) −ˆbNtNr
?
aˆb+1
(9)
whereˆb =
?
duni(R)
NtNr
, duni(R) is given in (8) and
ab=
?
1,
ab−1+ NtNrab−u,
1 ≤ b ≤ u
u < b ≤ˆb + 1.
(10)
Proof: Due to space constraints, only a proof sketch for
the discrete input case is given in the Appendix.
Theorem 1 characterises the tradeoff between outage diversity,
transmission rate and delay u for the MIMO block-fading
channel. When u = 0, it can be shown that d(R) = ∞, and
when u = B, Theorem 1 yields the outage diversity for the
uniform power allocation case (8). For systems with Gaussian
input constellations, the outage diversity d(R) is independent
of R, given by
d(R) = NtNr
B
?
b=1
ab,
(11)
where ab’s are defined in (10). The tradeoff between outage
diversity and CSIT delay is reflected through the recursive
function of ab in (10). For systems with discrete input con-
stellation, additional tradeoff between transmission rate R and
outage diversity is observed. Specifically, increasing R reduces
duni(R) andˆb in (9), thus significantly reducing d(R). When
R is sufficiently large, ab’s in (10) take value 1, and thus the
outage diversity d(R) is the same as that of the uniform power
allocation case. The rate threshold for zero-gain in outage
diversity is given in the following.
Corollary 1: Consider a MIMO block-fading channel with
causal CSIT and optimal power adaption (6). The outage
diversity is equal to that of uniform power allocation, i.e.,
d(R) = duni(R), if
duni(R)
NtNr
≤ u.
(12)
Corollary 1 provides an important rule-of-thumb for power
adaptation with causal CSIT. Whenever (12) is satisfied, CSIT
is useless in terms of improving the outage diversity. Hence,
uniform power allocation yields the optimal diversity.
For 0 < u <
(10), a linear homogeneous difference equation, which cannot
be solved in its general form, but for specific u can be solved
using standard techniques [12]. Special cases of interest are
considered in the following.
Corollary 2: Suppose u = 1 in Theorem 1, then
duni(R)
NtNr, (9) and (11) requires the solution to
d(R) = (1 + NtNr)ˆb?
whereˆb =
?duni(R)
Proof: With u = 1, it follows from (10) that
ab= (1 + NtNr)ab−1= (1 + NtNr)b−1,
1 +
?
duni(R) −ˆbNtNr
??
− 1, (13)
NtNr
?
.
b = 2,...,ˆb + 1.
The proof then follows from (9).
Noting that with discrete input constellation X of size 2M,
u = 1 and R ∈?0,MNt
BR in the first ARQ round, infinite feedback and a delay
constraint of B ARQ rounds, where each round is subject to
an i.i.d. Rayleigh flat fading channel matrix. Then, Corollary
2 is a special case of the results in [6,13], which derives the
optimal outage diversity of ARQ transmission with multi-bit
feedback over MIMO block-fading channels.
Corollary 3: Suppose u = 2 in Theorem 1, then
?ˆb+2
+duni(R) −ˆbNtNr
Δ2
B
?, the outage probability of the system
is equivalent to that of an ARQ system with transmission rate
d(R) = −1 +1
Δ
??1 + Δ
??1 + Δ
2
−
?1 − Δ
?ˆb+1
2
?ˆb+2?
−
?1 − Δ
2
?ˆb+1?
,
(14)
whereˆb =
Proof: With u = 2, the solution to (10) (which becomes
a 2nd order difference equation) is
??1 + Δ
Then, d(R) is readily obtained from (9).
Corollary 3 gives a closed-form expression to the outage
diversity for the special case u = 2. The SISO case Nt =
Nr= 1 simplifies further, where abin (10) forms a Fibonacci
series [17, p.381]. Therefore, noting thatˆb = duni(R) in SISO
channels, the outage diversity in (9) reduces to
?
duni(R)
NtNr
?
and Δ =√1 + 4NtNr.
ab=1
Δ2
?b
−
?1 − Δ
2
?b?
,b = 1,2,...
(15)
d(R) =
duni(R)
?
b=1
ξb= ξduni(R)+2− 1,
(16)
where ξnis the nthFibonacci number.
Fig. 1 illustrates how u affects the outage diversity for a
M = 4 bits/symbol modulation scheme with B = 8 blocks per
codeword. For systems with Gaussian input and a given delay
u, the maximum diversity is achieved for all rates. For discrete
input constellation, rate-diversity tradeoff is observed. For the
SISO case, Fig. 1 (left), we see that significant gains over
uniform power allocation are possible for low code rates, but
these gains rapidly decrease as the rate increases. For u = 4
corresponding to half a codeword delay in the transmitter
obtaining the CSI, there is no improvement in diversity for
rates above 2 bits per channel use. For the 2 × 2 MIMO
case, Fig. 1 (right), we see that although the outage diversity
is significantly affected by u (e.g. we see several orders of
magnitude difference between u = 1 and u = 2 cases), it is
so large that for all intents and purposes may be considered
infinite. Even when the the delay is half a codeword (u = 4),
for low code rates, power adaptation yields very large gains
over uniform allocation. Fig. 1 also shows that for a given
delay u, there is a threshold R at which power adaptation
gives no improvement in outage diversity over uniform power
allocation, as derived in Corollary 1.
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0 0.511.522.53 3.54
10
0
10
1
10
2
10
3
R (bits/channel use)
d(R)
u = 1
u = 2
u = 4
u = B
012345678
10
0
10
1
10
2
10
3
10
4
10
5
10
6
R (bits/channel use)
d(R)
u = 1
u = 4
u = B
u = 2
Fig. 1.The effect of u on outage diversity for an M = 4, B = 8 system: (left) SISO channel; (right) 2 × 2 MIMO channel.
V. CONCLUSIONS
In this paper we considered the MIMO block-fading channel
where the transmitter only has causal CSI. In particular, we
considered generalised causality, whereby at block b, the
transmitter only has knowledge of the channel fades of blocks
1,...,(b − u) for a fixed integer 0 ≤ u ≤ B. We derived a
general expression for obtaining the outage diversity of the
system, which uses optimal power adaptation subject to a
long-term power constraint. Our results generalise previous
works on block-fading channels with causal CSIT, which are
contained as special cases. In addition, we showed that for a
fixed rate, constellation size and number of codeword blocks,
there is a threshold CSIT delay at which power adaptation
no longer improves the outage diversity compared to uniform
power allocation. For SISO channels this threshold turns out
to be the Singleton bound (which is the outage diversity of
the system with no CSIT), i.e. for CSIT delays larger than the
Singleton bound, power adaptation gives no improvement in
outage diversity. Our results give new insight and important
design criteria for delay-limited systems where CSIT delay is
also a practical system constraint.
APPENDIX
The power constraint in (2) is written as
?
H(B)
B
?
b=1
tr
?
Pb
?
H(b−u)??
fH(B)
?
H(B)?
dH(B)≤ P.
Following the analysis in [18], define the normalised fading
gains ωb,t,r = −log |hb,t,r|2
are the entries of Hb. The fading gains ωb,t,r are collected
into a matrix Ωb ∈ RNr×Nt,b = 1,...,B. We also define
Ω(b)= diag(Ω1,...,Ωb). Further define πb
?
from H(B)to Ω(B),
log P
, where hb,t,r = |hb,t,r|eiθb,t,r
?
Ω(b−u)?
≡
πb
H(b−u)?
?
log tr(Pb(H(b−u)))
log P
. By changing the variable
B
?
b=1
?
Ω(b−u)∈A
Pπb(Ω(b−u))P−Pb−u
b?=1,t,rωb?,t,rdΩ(b−u)˙≤P,
where A
lemma [19, Sec. 4.3], πb
=
R(b−u)Nr×(b−u)Nt
+
?
b−u
?
. Applying Varadhan’s
such that1
Ω(b−u)?
sup
Ω(b−u)
⎧
⎨
⎩πb
?
Ω(b−u)?
−
b?=1,t,r
ωb?,t,r
⎫
⎬
⎭≤ 1,b = 1,...,B
asymptotically satisfies the power constraint. Since the outage
probability decreases with the transmit power, the optimal
power allocation rule satisfies
πb(Ω(b−u)) = 1 +
b−u
?
b?=1,t,r
ωb?,t,r,b = 1,...,B.
(17)
Hence, for large P, the optimal power allocation rule is
Pb
?
H(b−u)?.= Pπb(Ω(b−u))INt,
(18)
where INtis an identity matrix of size Nt× Nt. Therefore,
in the remainder of the proof, we consider
Pb
?
H(b−u)?
The proof is now divided into two parts. First, we prove
that the outage diversity d(R) is lower bounded by (9). Then,
we prove that (9) also upper bounds d(R).
= Pb
?
H(b−u)?
INt
.= Pπb(Ω(b−u))INt. (19)
A. Lower bound on outage diversity
From (5), for a given channel realization Hbin block b,
IX
??
Pb
?
Ω(b−u)?
Hb
?
.= MNt−
1
2MNt
?
x∈XNt
Ez
⎡
⎣log2
Pb−u
⎛
⎝
?
x?∈XNt
eT(x,x?,z)
⎞
⎠
⎤
⎦,
to
(20)
1For compact
PNt
t=1
notation,
b?=1,t,rωb?,t,r
isuseddenote
Pb−u
b?=1
PNr
r=1ωb?,t,r.
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where
T(x,x?,z) =
⎧
Nr
?
r=1
⎨
⎩−
?????
Nt
?
t=1
P
πb−ωb,t,r
2
eiθb,t,r(xt− x?
t) + zr
?????
2
+ |zr|2
⎫
⎬
⎭.
Let ωb,t= minr=1,...,Nr{ωb,t,r} and define
S(?)
For any ? > 0, in the limit for large P [6],
b
?
?
t ∈ {1,...,Nt} : πb(Ω(b−u)) − ωb,t> ?
?
.
(21)
IX
??
Pb
?
Ω(b−u)?
Hb
?
≥ M|S(?)
b|.
(22)
The outage probability therefore satisfies
Pout(P,R)˙≤Pr
?
Ω(B)∈ O(?)?
,
(23)
where, by letting S
(?)
b
denotes the complement of S(?)
B
?
b,
O(?)=
?
Ω(B)∈ RBNtNr:
b=1
???S
(?)
b
??? ≥duni(R)
Nr
?
.
(24)
From Varadhan’s lemma [19], the outage diversity satisfies
d(R) ≥ inf
Ω(B)∈O(?)∩RBNtNr
+
⎧
⎩
⎨
?
b,t,r
ωb,t,r
⎫
⎬
⎭.
(25)
Since the argument of the infimum in (25) is increasing with
ωb,t,r, it follows from (24) that the infimum is attained with
ωb,t,r=
?
πb
?
Ω(b−u)?
− ?,
(b − 1)Nt+ t ≤duni(R)
otherwise.
Nr
0,
=
⎧
⎪
⎪
⎩
⎨
1 − ?,
ωb−1,1,1+ NtNrωb−u,1,1,
0,
b ≤ u
b > u;(b − 1)Nt+ t ≤duni(R)
otherwise.
Nr
Letˆb =
?
duni(R)
NtNr
?
and ab= ωb,1,r, we have from (25) that
d(R) ≥ NtNr
ˆb
?
b=1
ab+ Nr(duni(R) −ˆbNt)aˆb+1,
(26)
where
ab=
?
1 − ?,
ab−1+ NtNrab−u,
1 ≤ b ≤ u
u < b ≤ˆb + 1.
(27)
By letting ? ↓ 0, the outage diversity is lower bounded by (9).
B. Upper bound on outage diversity
The input-output mutual information of a MIMO channel
is upper bounded by that of a genie aided receiver, where
the interference caused by multiple transmit antenna is com-
pletely removed at the receiver. The resulting channel therefore
consists of Nt parallel channels, each component channel
consists of one transmit and Nrreceived antenna. Therefore,
IX
??
Pb(H(b−u))Hb
?
?Nr
≤
Nt
?
t=1
IX
??
Pb
?
H(b−u)?
Letting
ξb,t
?
,
where
min{ωb,t,r,r = 1,...,Nr}, it follows that ξb,t
Thus, for any ? > 0, in the limit for large P,
ξb,t
=
r=1|hb,t,r|2.
ωb,t
.= P−ωb,t.
=
IX
??
Pb(H(b−u))Hb
?
≤
Nt
?
t=1
IX
?
P
πb(Ω(b−u))−ωb,t
2
?
≤ M|S(−?)
b
|,
where S(−?)
from part A of the proof that the outage diversity is upper
bounded by (9).
b
is defined by (21). By letting ? ↓ 0, it then follows
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