Fundamental diversity, multiplexing, and array gain tradeoff under different MIMO channel models

Abstract

Following the seminal work of Zheng and Tse on the diversity and multiplexing tradeoff (DMT) of MIMO channels, in this paper we introduce the array gain to investigate the fundamental relation between transmission rate and reliability in MIMO systems. The array gain gives information on the power offset that results from exploiting channel state information at the transmitter or as a consequence of the channel model. Hence, the diversity, multiplexing, and array gain (DMA) analysis can be directly translated into a parameterized characterization of its associated outage probability performance. In this paper we derive the fundamental DMA tradeoff achievable by any scheme in uncorrelated Rayleigh, semicorrelated Rayleigh, and uncorrelated Rician block-fading MIMO channels. We use these results to analyze the effect of important channel parameters in the outage performance at different points of the DMT curve.

Full text

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FUNDAMENTAL DIVERSITY, MULTIPLEXING, AND ARRAY GAIN TRADEOFF
UNDER DIFFERENT MIMO CHANNEL MODELS
Luis G. Ord´ o˜ nez,1Daniel P. Palomar,2and Javier R. Fonollosa1
1Signal Processing and Communications Group, Universitat Polit` ecnica de Catalunya - Barcelona Tech
e-mail:{luis.g.ordonez, javier.fonollosa}@.upc.edu
2Dept. of Electric and Computer Engineering, Hong Kong University of Science and Technology (HKUST)
e-mail: palomar@ust.hk
ABSTRACT
Following the seminal work of Zheng and Tse on the diversity and
multiplexing tradeoff (DMT) of MIMO channels, in this paper we
introduce the array gain to investigate the fundamental relation be-
tween transmission rate and reliability in MIMO systems. The array
gain gives information on the power offset that results from exploit-
ing channel state information at the transmitter or as a consequence
of the channel model. Hence, the diversity, multiplexing, and array
gain (DMA) analysis can be directly translated into a parameterized
characterization of its associated outage probability performance. In
this paper we derive the fundamental DMA tradeoff achievable by
any scheme in uncorrelated Rayleigh, semicorrelated Rayleigh, and
uncorrelated Rician block-fading MIMO channels. We use these re-
sults to analyze the effect of important channel parameters in the
outage performance at different points of the DMT curve.
1. INTRODUCTION
Multiple-input multiple-output (MIMO) channels are well know to
provide a number of advantages over conventional single-antenna
(SISO) channels, which have been traditionally described by the di-
versity, multiplexing, and array gains [1].
ZhengandTseshowedintheexcellentgroundbreakingpaper[2]
that both diversity and multiplexing gains can be simultaneously ob-
tained, but there is a tradeoff between how much of each type of gain
any MIMO scheme can extract: higher spatial multiplexing comes at
the price of sacrificing diversity. To be more specific, [2] focuses on
the high-SNR outage probability and derives the fundamental trade-
off curve achievable by any scheme, where the spatial multiplexing
gain is understood as the fraction of capacity attained at high SNR
and the diversity gain indicates the high-SNR reliability of the sys-
tem. The main problem, however, is that the diversity and multiplex-
ing tradeoff (DMT) provides only a coarse measure of performance
in the sense that it does not capture the impact of various relevant
channel features and it is also insensitive to the presence of channel
state information (CSI) at the transmitter. Furthermore, it is difficult
to translate any conclusion extracted from the DMT into the actual
error probability of a particular scheme.
Several attempts have been made in the literature to provide the
DMT with operational significance. First, we have the finite-SNR
DMT derived in [3], where diversity and multiplexing gains defi-
nitions are modified to hold for any finite SNR value. However,
1This work was partly funded by the project 2009SGR1236 of the Cata-
lan Administration, and by the Spanish Science and Technology Commis-
sions and FEDER funds from the EC (TEC2006-06481, TEC2010-19171,
and CONSOLIDER INGENIO CSD2008-00010 COMONSENS).
2This work was funded by the Hong Kong RGC 618709 research grant.
the derivations are based on a lower bound on the outage probabil-
ity and the final results require an additional numerical optimization
process. Similarly, the finite-SNR DMT is also addressed in [4] for
asymptotically large systems as either the number of antennas in one
or both sides of the link tends to infinity. A totally different approach
is taken in [5], where the focus is again on the high-SNR regime but
the notion of multiplexing gain is substituted by that of rate region
to investigate scenarios in which the data rate does not scale linearly
withthelogarithmoftheSNRasin[2]. However, thethroughputand
reliability tradeoff in [5] is still independent of important parameters
of the channel model.
Here we aim at completing the DMT framework by including
the array gain in the picture while trying to keep the essence of
the original formulation. That is, we use equivalent definitions of
the diversity and multiplexing gains to those in [2] and include a
new performance indicator which is able to cope with the limita-
tions of the DMT. The array gain, indeed, gives information on the
power offset that results from exploiting CSI at the transmitter or as a
consequence of the adopted channel model. The resulting diversity,
multiplexing, and array gain (DMA) analysis provides then more in-
sights into the fundamental relation between transmission rate and
reliability in MIMO systems, since the error probability is now char-
acterized by two parameters: the diversity and array the gain. In
this sense, the DMA tradeoff is still a two-fold tradeoff and must
be not understood as a three-sided compromise between diversity,
multiplexing, and array gains.
As a first step towards this objective, the fundamental DMA
gains achievable by any scheme in uncorrelated Rayleigh block-
fading MIMO channels were derived in [6]. Here, we complete this
result and derive the fundamental DMA gains under semicorrelated
Rayleigh and uncorrelated Rician fading. This allows us to investi-
gate the influence of the channel model in the outage probability per-
formance. The rest of the paper is organized as follows. Sections 2.1
and 2.2 introduce the channel and system models. Then, the DMA
tradeoff framework is formulated in Section 2.3 and derived in Sec-
tion 3. Finally, the main contribution of the paper is summarized in
Section 4.
2. PRELIMINARIES AND PROBLEM FORMULATION
2.1. Channel Model
A MIMO channel with nTtransmit and nRreceive dimensions can
be described by an nR× nT channel matrix H, whose (i,j)then-
try characterizes the propagation path between the jthtransmit and
the ithreceive antenna. Usually, since there are a large number of
scatters in the channel that contributes to the signal at the receiver,
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the application of the central limit theorem results in Gaussian dis-
tributed channel matrix coefficients. Analogously to the single an-
tenna channel, this model is referred to as MIMO Rayleigh or Rician
fading channel, depending on whether the channel entries are zero-
mean or not. Here we consider the following important particular
cases of this general channel model:
Definition i. The uncorrelated Rayleigh MIMO fading channel
model is defined as
H = Hw
where Hw is a nR× nT random matrix with i.i.d. zero-mean unit-
variance complex Gaussian entries.
(1)
Definition ii. The semicorrelated Rayleigh fading MIMO channel
model with correlation at the side with minimum number of anten-
nas1is defined as
?
HwΣ1/2
where Σ =?Σ1/2??Σ1/2?†is the n×n positive definite correlation
··· > σn > 0, n = min(nT,nR), and Hwis defined in (1).
Definition iii. The uncorrelatedRician fading MIMOchannel model
is defined as
?
where K ∈ (0,∞) is the Rician factor, Hlosis a nR× nTdetermin-
istic matrix containing the line-of-sight components of the channel,
and Hw is defined in (1). For convenience, we introduce the non-
centrality matrix Ω defined as
?
H =
Σ1/2Hw
nR≤ nT
nR> nT
(2)
matrix with eigenvalues σ = (σ1,...,σn) ordered such that σ1 >
H =
K
K + 1Hlos+
?
1
K + 1Hw
(3)
Ω =
KHlosH†
KH†
losHlos
los
nR≤ nT
nR> nT
(4)
with eigenvalues ω = (ω1,...,ωn) ordered such that ω1 > ··· >
ωn > 0.
2.2. System Model
We consider a wireless communication system with nTtransmit and
nRreceive antennas, in which the channel matrix H is drawn from
one of the channel models presented in Section 2.1 and remains con-
stant within a block of nSsymbols after which it changes to an inde-
pendent realization. In this situation, the received signal within one
block can be gathered in an nR×nSmatrix Y related to the nT×nS
transmitted matrix X as
Y = HX + W
(5)
where W is the additive white Gaussian noise and has i.i.d. entries
with zero mean and unit variance. The transmitted signal X is nor-
malized forcing the transmit power per block to satisfy
1
nSE??X?2
F
?≤ snr
(6)
1When, in contrast to Definition ii, we have correlation at the side with
maximum number of antennas, the joint distribution of the channel eigen-
values is slightly different, but the proof techniques employed here for the
min-semicorrelated channel model apply verbatim to the max-semicorrelated
channel model.
where snr is the average SNR per receive antenna. In addition, the
instantaneous CSI is assumed to be perfectly known at the receiver
but may or may not be available at the transmitter.
Under this system setup, the outage probability is the primary
measure of interest in the sense that it is widely accepted to be the
best achievable block error probability (BLER) in the limit of large
codewordlength. Theoutageprobabilityisdefinedastheinfimumof
the probability that the instantaneous mutual information falls below
the transmission rate R and, for the system model in (5), is given
by [2, Sec. III.B]
?
Pout(R,snr) = inf
Q≥0,tr(Q)≤snrPr
logdet(InR+ HQH†) < R
?
(7)
.
2.3. DMA Tradeoff Formulation
Recall that our main objective is to complete the DMT framework by
including the array gain. Hence, let us first formalize the concepts
of diversity, multiplexing, and array gain.
As in [2], we define a scheme as a family of codes {C(snr)} of
block length nS, which employs a different code C(snr) with rate
R(snr) for each SNR level. Then, a MIMO coding scheme {C(snr)}
is said to achieve a spatial multiplexing gain r, a diversity gain d(r),
and an array gain a(r) if the data rate is such that
lim
snr→∞
R(snr)
logsnr= r
(8)
with 0 ≤ r ≤ min(nT,nR) and the outage probability satisfies
logPout(r,snr)
logsnr
Pout(r,snr)
snr−d(r)
lim
snr→∞
= −d(r)
(9)
lim
snr→∞
= a(r)−d(r).
(10)
Observe that definitions in (9) and (10) induce the following
approximation of the high-SNR behavior of the outage probability
when R satisfies (8):
Pout(r,snr) ∼?a(r) · snr?−d(r)
where ‘∼’ denotes asymptotic equivalence as snr → ∞. Hence, the
DMA analysis of a particular system can be directly translated into
a parameterized characterization of its associated outage probability
performance as opposed to the DMT, which only provides the slope
of the outage probability curve for a given multiplexing gain. This
enables the direct comparison of different strategies under different
channel models and CSI assumptions.
(11)
3. DMA TRADEOFF IN MIMO CHANNELS
In this section we provide the fundamental DMA analysis for the
channel models introduced in Section 2.1. For technical reasons,
we restrict our attention to the case in which the transmit covariance
matrix Q is a scaled identity matrix, i.e.,
Q =snr
φInT
(12)
being φ a strictly positive constant. This suffices, however, to in-
clude the most interesting scenarios when CSI is available both at
the transmitter and receiver for φ = min(nT,nR) or only at the re-
ceiver for φ = nT(see [6, Sec. IV] for details).
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Given the transmit covariance matrix in (12) and assuming that
R satisfies (8), the high-SNR behavior of the outage probability is
given by
P(φ)
out(r,snr) ∼ Pr
?
n
?
t=1
?1 +snr
φλt
?< snrr?
(13)
=?a(φ)(r) · snr?−d(r)+ o?snr−d(r)?.
(14)
where n = min(nT,nR) and λ1 ≥ ··· ≥ λn ≥ 0 are the non-
zero ordered eigenvalues of HH†. From (13) it is clear that the
fundamental DMA gains under the channel models in Definitions i,
ii, and iii follow from obtaining the first order series expansion of
the outage probability. It is well known, however, that these channel
models have the same fundamental DMT as the one for uncorrelated
Rayleigh MIMO channels derived in [2, Thm. 1]. Hence, we can al-
ready present the diversity gain d(r) as a function of the multiplexing
gain r in the next lemma.
Lemma 1 [7]. The diversity gain d(r) in a nR× nTMIMO system
with multiplexing gain 0 ≤ r ≤ n and the transmit covariance ma-
trix as in (12) is given for the channel models in Definitions i, ii, and
iii by
d(r) = Gd(k) − Gr(k)r
(15)
where k = ?r? = {maxk ∈ N|k ≤ r}, and
Gd(k) = mn − k(k + 1)
Gr(k) = m + n − (2k + 1)
with n = min(nT,nR) and m = max(nT,nR).
(16)
(17)
Now, in order to complete the fundamental DMT analysis, we
only need to obtain the array gain as done in next theorem.
Theorem 1 [8]. The array gain a(φ)(r) in a nR× nTMIMO system
with multiplexing gain 0 < r < n and the transmit covariance as in
(12) is given by
i. Uncorrelated Rayleigh Fading (see Definition i):
?
where
a(φ)
i
(r) =Ki(k)|A(k)|
?φGd(k)
?k
Gr(k)
??−1/d(r)
(18)
Ki(k) =
t=1t!
?n−k
t=1(k + t − 1)!?n
?
t=1(m − t)!
(19)
matrix A(k) ((n − k − 1) × (n − k − 1)) is defined as
m−n
?
and the rest of parameters are as given in Lemma 1.
[A(k)]u,v =
t=0
m − n
t
?
(−1)t
(u + v + t)
(20)
ii. Semicorrelated Rayleigh Fading (see Definition ii):
?
where
a(φ)
ii(r) = Kii(k,σ)|Aii(k,σ)||A(k)|
?φGd(k)
Gr(k)
??−1/d(r)
(21)
Kii(k,σ) =
(−1)k(n−k)
?n
t=1σm
t(m − t)!
n
?
i<j
?σiσj
σj− σi
?
(22)
and matrix Aii(k,σ) (n × n) is defined as
[Aii(k,σ)]u,v =
⎧
⎪
=
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
1
(n−k−v−1)!
(v − (n − k))!?1
(v−k−1)!
?−1
σu
?(n−k)−v−1ln(σu)
1 ≤ v ≤ min(n − k − 1,k)
?(n−k)−v−1
σu
n − k ≤ v ≤ k
k < v ≤ n
1
?−1
σu
?v−k−1
. (23)
iii. Uncorrelated Rician Fading (see Definition iii):
a(φ)
iii(r) =
?
Kiii(k,K,ω)|Aiii(k,ω)||A(k)|
?φGd(k)
Gr(k)
??−1/d(r)
(24)
where
Kiii(k,K,ω) = (−1)k(n−k)e−mnK(K + 1)Gd(k)
?n
i<j(ωj− ωi)
(25)
and matrix Aiii(k,ω) (n × n) is defined as
?
[Aiii(k,ω)]u,v =
v!
(m−k)!(n−k)!?v(k,ωu)
1
(v−k−1)!(m−n+v−k−1)!ωv−k−1
1 ≤ v ≤ k
k < v ≤ n
u
(26)
where ?v(k,ωu) = ωn−k
with2F2(·;·) denoting the generalized hypergeometric function [9,
eq. (9.14.1)].
u
2F2(v +1,1;m−k +1,n−k +1;ωu)
The channel models considered in this paper offer equal diver-
sity gain but different array gain. Hence, in contrast to the DMT,
the DMA analysis elucidates the performance gap between them.
Choosing the uncorrelated Rayleigh fading channel as the reference
channel, we define the asymptotic SNR gap as
Δj(r) = 10log10
?a(φ)
j
(r)
a(φ)
i
(r)
?
dB (27)
where a(φ)
Definition j with j ∈ {ii,iii}. Observe that a positive Δj(r) in-
dicates that the uncorrelated Rayleigh outage probability is outper-
formed, whereas a negative Δj(r) implies the opposite situation. In
other words, Δj(r) indicates in how many dB’s we have to increase
(Δj(r) < 0) or reduce (Δj(r) > 0) the nominal SNR to achieve the
same outage probability as in the uncorrelated Rayleigh case.
j
(r) denotes the array gain under the channel model in
Corollary 1. The asymptotic SNR gap Δj(r) at a multiplexing gain
point 0 < r < n with respect to the uncorrelated Rayleigh channel
model is given by
ii. Semicorrelated Rayleigh Fading (see Definition ii):
Δii(r) =
10
d(r)log10
?
Ki(k)
Kii(k,σ)|Aii(k,σ)|
?
dB.
(28)
iii. Uncorrelated Rician Fading (see Definition iii):
Δiii(r) =
10
d(r)log10
?
Ki(k)
Kiii(k,K,ω)|Aiii(k,ω)|
?
dB.
(29)
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snr
Pout(r,snr)
r = 0.8
r = 1.5
Δiii(r)
Δii(r)
0
10
2030
40
50
60
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
snr
Pout(r,snr)
r = 1.8
r = 2.2
Δiii(r)
Δii(r)
0
20 40
60
80
100
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Fig. 1. Simulated outage probability (solid) and DMA analysis (dashed) in a 3 × 3 MIMO system under i. uncorrelated Rayleigh fading
(blue), ii. semicorrelated Rayleigh fading (red), and iii. uncorrelated Rician fading (green).
For illustrative purposes we show in Figure 1 the simulated
outage probability and the high-SNR outage probability charac-
terization derived from the DMA tradeoff with an isotropic in-
put covariance matrix and for the three addressed channel mod-
els. For the semicorrelated Rayleigh channel we have set σ =
(2.2648,0.5100,0.2252) and for the uncorrelated Rician channel
we have chosen K = 1 and ω = (6.3638,2.4319,0.2043). Since
the target data rate is chosen as R = rlogsnr, the simulated outage
probability is not representative for low SNR values. The remaining
part of the curve is, as expected, well approximated by the DMA
analysis. We emphasize that the traditional DMT (Lemma 1) pro-
vides the slopes of the curves but not the horizontal shift, which is
precisely the contribution of the present paper (Theorem 1).
In addition, we can observe in Figure 1 that the SNR gap for a
fixed performance under different channel models is approximately
constant in the range of outage probabilities of interest. Hence, the
asymptotic SNR gap given in Corollary 1 provides a good prediction
on the performance degradation/gain due to the channel model. This
fact is further confirmed in Table 1, where we compare the simu-
lated SNR gap at an outage probability of 10−8and the asymptotic
SNR gap for different system setups. The difference between both
measures is in most cases smaller than 0.2 dB.
4. CONCLUSIONS
Zheng and Tse 2003 paper was the first one to reveal and quantify
the fundamental interconnection present in MIMO channels between
the multiplexing gain, associated to rate, and the diversity gain, re-
lated to the slope of the error rate. This characterization is however
difficult to be translated into practical performance indicators like
˜Δii(r)Δii(r)
˜Δiii(r)Δiii(r)
r = 0.8
r = 1.5
r = 1.8
r = 2.2
-3.89 dB
-4.99 dB
-7.02 dB
-7.31 dB
-3.83 dB
-5.07 dB
-7.92 dB
-7.31 dB
2.21 dB
2.41 dB
3.40 dB
1.27 dB
2.40 dB
2.37 dB
3.27 dB
1.27 dB
Table 1. Simulated SNR gap˜Δj(r) at Pout = 10−8and asymp-
totic SNR gap Δj(r) under ii. semicorrelated Rayleigh fading and
iii. uncorrelated Rician fading.
outage probability without the addition of a third parameter, the ar-
ray gain, that provides the shift to the diversity gain slope resulting
in an asymptotic affine characterization of the error curve in the log-
arithmic domain. This paper introduces the DMA analysis under
uncorrelated Rayleigh, semicorrelated Rayleigh, and uncorrelated
Rician MIMO channels, opening the door for a more illustrative per-
formance evaluation of MIMO schemes.
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