Outage Probability Conjecture Does Not Hold for
Two-Input-Multiple-Output (TIMO) System
Gen Li, Jingkai Yan, and Yuantao Gu
Beijing National Research Center for Information Science and Technology (BNRist)
Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
Email: {g-li16, yjk14}@mails.tsinghua.edu.cn, gyt@tsinghua.edu.cn
Abstract—Multiple-Input-Multiple-Output (MIMO) communi-
cation systems have seen wide application due to its performance
benefits such as multiplexing gain. For MIMO systems with
non-ergodic Gaussian channel, a conjecture regarding its outage
probability has been proposed by Telatar in [1] and long
considered true. A special single-output case of this conjecture
has been proved theoretically. In this work, we address the special
Two-Input-Multiple-Output (TIMO) case, and show that the
conjecture is untrue. A concrete counter-example is proposed and
verified both theoretically and by numerical experiments. This
result rejects the decades-long conjecture and provides interesting
insight into the symmetry of MIMO systems.
I. INTRODUCTION
Multi-Input-Multi-Output (MIMO) have been one of the
core concepts and practices in modern communication systems
since its pioneering proposal in the 1990s [2], [3]. With
multiple transmitters and receivers, data rate and multi-user
capabilities can be improved significantly [4], [5]. Current
and future generations of communication systems all embrace
MIMO as an essential part, including WLAN, LTE, 5G,
and mmWave MIMO systems [6]–[10]. [11] reviews on the
importance of MIMO in contemporary communication.
The theoretical channel capacity of MIMO systems is be-
lieved to be first addressed by Telatar in his work [1]. Capacity
for mean (ergodic) Gaussian channel was solved by that work,
and some later works addressed problems including parameter
setting [12] and multi-user interference [13]. However, most
existing works only address the channel capacity problem for
mean (ergodic) Gaussian channels, and few works exist on the
no less important non-ergodic channel [1]. For the capacity
of non-ergodic MIMO channels, the author in [1] adopted a
different approach as the outage probability, and proposed a
conjecture on conditions for the minimum outage probability.
This conjecture has long been assumed true and employed for
the benchmark outage probability in later works [14], [15].
Proof of a special Multiple-Input-Single-Output (MISO) case
of the conjecture was established in [15].
This paper approaches the Outage Probability Conjecture
from a novel perspective, and provides a theoretical analysis
of the particular Two-Input-Multiple-Output (TIMO) case. Our
This work was partially supported by National Natural Science Foun-
dation of China (NSFC 61571263, 61531166005), the National Key Re-
search and Development Program of China (Project No. 2016YFE0201900,
2017YFC0403600), and Tsinghua University Initiative Scientific Research
Program (Grant 2014Z01005). The corresponding author of this work is Y. Gu.
analysis reveals that the conjecture is actually false, with
a concrete counter-example provided and verified by both
theoretical derivation and numerical simulation
1
, providing
interesting insight into the symmetry of Gaussian MIMO
channels.
The rest of this paper is organized as follows. Section II
reviews the Outage Probability Conjecture and the proven
case of MISO. Theoretical analysis of the TIMO case and the
counter-example is given in Section III, which is the main con-
tribution of our work. Numerical experiments are conducted
in Section IV. Section V contains concluding remarks.
Notations: H
T
and H
⇤
denote respectively the transpose
and conjugate transpose of H. #S denotes the number of
elements in set S.
II. T
HE OUTAGE PRO BAB IL IT Y CONJECTURE
A. The Conjecture
Consider a single-user MIMO Gaussian channel. Assume
the number of transmitting antennas and receiving antennas is
t and r, respectively. The channel model can be described as
y = Hx + n, (1)
where x 2 C
t
, y 2 C
r
, and n 2 C
r
denotes the transmitted
vector, the received vector, and the additive noise, respectively.
The channel is denoted by H 2 C
r⇥t
, where each entry
is i.i.d. circularly symmetric complex Gaussian variable with
E |H
i,j
|
2
=1.
In the non-ergodic case, the matrix H is random but is
held fix once it is chosen. In this case, we consider the outage
probability for evaluation of the channel. Let R be the data rate
and P be the signal power constraint. The outage probability
P
out
(R, P) is defined as follows:
P
out
(R, P):= inf
Q:Q0,
tr(Q)P
P[log det(I
r
+ HQH
⇤
) <R]. (2)
In Telatar’s words, P
out
(R, P) is a probability such that “For
any rate less than R and any there exists a code satisfying
the power constraint P for which the error probability is less
than for all but a set of H whose total probability is less
than P
out
(R, P).” The conjecture is stated as the following
[1]:
1
The supplementary downloadable material, including MATLAB codes for
all experiments, is available at http://gu.ee.tsinghua.edu.cn/publications/.