68 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 1, JANUARY 2011
On the Diversity Gain in MIMO Channels with
Joint Rate and Power Control Based on Noisy CSITR
Xiao Juan Zhang, Yi Gong, Senior Member, IEEE, and Khaled Ben Letaief, Fellow, IEEE
Abstract—We analyze the impact of imperfect channel state
information at the transmitter and receiver (CSITR) on the
achievable diversity gain in multi-input multi-output (MIMO)
fading channels with joint rate and power control. With rate
control only, we consider the system with and without a minimum
spatial multiplexing gain constraint, respectively. With joint rate
and power control, we show that the achievable diversity gain
canbeimprovedsignificantly. The conducted analysis adopts the
diversity-multiplexing tradeoff framework and uses the notions
of diversity gain and multiplexing gain to study the impact of
noisy CSITR in MIMO fading channels.
Index Terms—Diversity-multiplexing tradeoff, joint rate and
power control, diversity gain, multiplexing gain, MIMO.
I. INTRODUCTION
IN multi-input multi-output (MIMO) fading channels, the
diversity-multiplexing (D-M) tradeoff framework proposed
by Zheng and Tse in [1] has proved to be very useful in
understanding how diversity and rate interplay in the high
signal-to-noise-ratio (SNR) regime. While many relevant con-
tributions in literature have assumed that perfect channel state
information (CSI) at the receiver (CSIR) is known but CSI
at the transmitter (CSIT) is not, it is not surprising that the
D-M tradeoff can be further enhanced through power and/or
rate control with the aid of CSIT [2]-[4]. If the CSIT is
perfectly known, the probability of outage occurrences can be
significantly reduced in interference-free slow fading channels
since the transmitter is always able to adjust its power or rate
adaptively according to the instantaneous channel conditions.
In practice, the CSIT is almost always imperfect due to
quantized CSI feedback in frequency division duplex (FDD)
systems or imperfect channel estimation at the transmitter in
time division duplex (TDD) systems. For FDD systems, it is
shown in [5] and [6] that quantized channel feedback leads
to a substantial diversity improvement with power control
in MIMO channels. Joint rate and power control has also
been considered, based on which the optimal D-M tradeoff
is derived for adaptive-rate FDD systems [6]. As for TDD
systems, it is shown in [7]-[9] that the achievable diversity
gain can be significantly improved with power control based
on noisy CSIT in multi-antenna channels, while the work in
Manuscript received May 14, 2010; revised July 25, 2010; accepted
September 4, 2010. The associate editor coordinating the review of this paper
and approving it for publication was L. Lampe.
X. J. Zhang and Yi Gong are with the School of Electrical and Elec-
tronic Engineering, Nanyang Technological University, Singapore (e-mail:
{zh0012an, eygong}@ntu.edu.sg).
K. B. Letaief is with the Department of Electronic and Computer Engineer-
ing, Hong Kong University of Science and Technology, Hong Kong (e-mail:
eekhaled@ece.ust.hk).
Digital Object Identifier 10.1109/TWC.2010.110310.100818
[10] shows that rate control can also improve the achievable
diversity gain under similar channel conditions.
In this work, we complement the literature of D-M tradeoff
analysis for TDD MIMO channels by considering rate control
based on noisy CSIT and CSIR (i.e., noisy CSITR). Effective
rate control schemes are proposed. Together with our power
control scheme in [7], joint rate and power control is also
considered. It is interesting to find that if there is no minimum
spatial multiplexing gain constraint 𝑟𝑚𝑖𝑛, rate control itself is
already able to boost the diversity gain unboundedly. When
there is a nonzero 𝑟𝑚𝑖𝑛, the diversity gain is limited by the
achievable diversity gain at 𝑟𝑚𝑖𝑛. In particular, it can be
retained at the diversity gain for 𝑟𝑚𝑖𝑛 in the entire range
of possible average multiplexing gains. When joint rate and
power control is carried out, it is shown that the D-M tradeoff
can be further improved significantly.
Notations: ℛ𝑁denotes the set of real 𝑁-tuples, and ℛ𝑁+
denotes the set of non-negative 𝑁-tuples. Likewise, 𝒞𝑁×𝑀
denotes the set of complex 𝑁×𝑀matrices. For a real number
𝑥,(𝑥)+denotes max(𝑥, 0), while for a set 𝒪⊆ℛ
𝑁,𝒪+
denotes 𝒪∩ℛ
𝑁+.𝒞𝒩(0,𝜎
2)denotes the complex Gaussian
distribution with mean 0 and variance 𝜎2. The superscript †de-
notes conjugate transpose, ∥⋅∥𝐹denotes the matrix Frobenius
norm, and I𝑁denotes the 𝑁×𝑁identity matrix. 𝐸{⋅} denotes
the expectation operator, log(⋅)denotes the base-2 logarithm,
and 𝒫(𝒜)denotes the probability of event 𝒜.𝑓(𝜌).
=𝜌𝑏
denotes that 𝑓(𝜌)is exponentially equal to 𝜌𝑏and that 𝑏is the
exponential order of 𝑓(𝜌), i.e., lim𝜌→∞ log 𝑓(𝜌)/log(𝜌)=𝑏.
II. SYSTEM MODEL
We consider a point-to-point TDD wireless link with 𝑀
transmit and 𝑁receive antennas, where the downlink and
uplink channels are reciprocal. Without loss of generality, we
assume 𝑀≥𝑁. We consider quasi-static Rayleigh fading
channels, where the channel gains are constant within one
transmission block of 𝐿symbols, but change independently
from one block to another. We assume that the channel gains
are independently complex circular symmetric Gaussian with
zero mean and unit variance. The channel model, within one
block, can be written as
Y=𝑃/𝑀HX +W(1)
where H={ℎ𝑛,𝑚}∈𝒞
𝑁×𝑀with ℎ𝑛,𝑚,𝑛=1,...,𝑁,
𝑚=1,...,𝑀, being the channel gain from the 𝑚-th transmit
antenna to the 𝑛-th receive antenna; X={𝑋𝑚,𝑙 }∈𝒞
𝑀×𝐿
with 𝑋𝑚,𝑙,𝑚=1,...,𝑀,𝑙=1,...,𝐿, being the sym-
bol transmitted from the 𝑚-th transmit antenna at time 𝑙;
Y={𝑌𝑛,𝑙}∈𝒞
𝑁×𝐿with 𝑌𝑛,𝑙,𝑛=1,...,𝑁,𝑙=1,...,𝐿,
being the signal received from the 𝑛-th receive antenna at
1536-1276/11$25.00 c
⃝2011 IEEE
ZHANG et al.: ON THE DIVERSITY GAIN IN MIMO CHANNELS WITH JOINT RATE AND POWER CONTROL BASED ON NOISY CSITR 69
time 𝑙; the additive noise W∈𝒞
𝑁×𝐿has independent and
identically distributed (i.i.d.) entries 𝑊𝑛,𝑙 ∼𝒞𝒩(0,𝜎
2);𝑃is
the instantaneous transmit power while the average energy of
𝑋𝑚,𝑙 is normalized to be 1. Letting ¯
𝑃denote the average sum
power constraint, we have 𝐸{𝑃}=¯
𝑃and the average SNR
at the receive antenna is 𝜌=¯
𝑃/𝜎2.
We assume that neither the receiver nor the transmitter has
any CSI initially. Before data transmission, two-way training is
performed to obtain channel knowledge, which consists of two
phases. In the first phase, the receiver sends training symbols
to the transmitter through the backward training channel. The
transmitter obtains noisy CSIT ˆ
H𝑏∈𝒞
𝑁×𝑀from the training
symbols using MMSE channel estimation. Thus, ˆ
H𝑏can be
modeled as [10], [11]
H=ˆ
H𝑏+E𝑏(2)
where the estimation error E𝑏∈𝒞
𝑁×𝑀has i.i.d. entries
𝐸𝑏,𝑛,𝑚 ∼𝒞𝒩(0,𝜎
2
𝑒,𝑏)and is independent of ˆ
H𝑏. The quality
of ˆ
H𝑏is thus characterized by 𝜎2
𝑒,𝑏.If𝜎2
𝑒,𝑏 =0, the transmitter
has perfect CSIT; if 𝜎2
𝑒,𝑏 increases, the transmitter has less
reliable CSIT. We follow [10] to quantify the channel quality
at the transmitter. The transmitter is said to have a CSIT quality
𝛼,if𝜎2
𝑒,𝑏
.
=𝜌−𝛼. The definition of 𝛼builds up a connection
between the imperfect CSIT and the forward channel SNR, 𝜌.
Since the variance of the channel estimation error is inversely
proportional to the training symbols’ SNR, any value of 𝛼can
be achieved by scaling the backward channel power.
In the second training phase, the transmitter sends training
symbols with power 𝑃𝑓to the receiver in the forward channel.
The receiver estimates the channel and obtains ˆ
H𝑓with
MMSE estimation. Thus, we have
H=ˆ
H𝑓+E𝑓(3)
where the estimation error E𝑓∈𝒞
𝑁×𝑀has i.i.d. entries
𝐸𝑓,𝑛,𝑚 ∼𝒞𝒩(0,𝜎
2
𝑒,𝑓 )and is independent of ˆ
H𝑓. It is reason-
able to let the transmitter spend the same amount of power
sending data symbols and training symbols in the forward
channel, i.e., 𝑃𝑓=𝑃. Therefore, we have 𝜎2
𝑒,𝑓 ∝𝑃−1.
III. D-M TRADEOFF ANALYSIS
For a MIMO scheme realized by a family of codes
{𝐶(𝜌)} with SNR 𝜌,rate𝑅(𝜌)(bits per channel use), and
maximum-likelihood (ML) error probability 𝒫𝑒(𝜌), Zheng
and Tse defined in [1] the spatial multiplexing gain 𝑟as
𝑟≜lim𝜌→∞ 𝑅(𝜌)/log 𝜌and the diversity gain 𝑑as 𝑑≜
−lim𝜌→∞ 𝒫𝑒(𝜌)/log 𝜌, and derived the optimal D-M trade-
off, hereby denoted as 𝑑𝑛𝑜𝑃 𝐶 (𝑟), for i.i.d. quasi-static flat
Rayleigh fading channels with perfect CSIR and no CSIT.
In particular, 𝑑𝑛𝑜𝑃𝐶 (𝑘)=(𝑀−𝑘)(𝑁−𝑘)for 𝑘=0, ..., 𝑁 .
With the use of capacity-achieving codes, the ML error
probability 𝒫𝑒(𝜌)of the channel described in (1) is dominated
by the corresponding outage probability 𝒫𝑜𝑢𝑡 . We will thus
leverage on the outage probability to examine the diversity
gain. Based on the noisy ˆ
H𝑏, the transmitter performs power
control 𝑃(ˆ
H𝑏)and rate control 𝑅(ˆ
H𝑏)for data transmission.
Note that the power control is subject to 𝐸{𝑃(ˆ
H𝑏)}=¯
𝑃
and the rate control is subject to 𝐸{𝑟(ˆ
H𝑏)}=¯𝑟,where
𝑟(ˆ
H𝑏)≜lim𝜌→∞ 𝑅(ˆ
H𝑏)/log 𝜌. We assume that the receiver
is able to know the data transmission rate for decoding. In
modern wireless systems, there is a common setup to inform
the receiver of transmission rate, e.g., in WiMAX [12], there is
a downlink signaling channel to indicate the downlink payload
transmission mode.
The channel model in (1) can be rewritten as
Y=𝑃(ˆ
H𝑏)/𝑀 (ˆ
H𝑓+E𝑓)X+W.(4)
The mutual information of this channel is difficult to com-
pute since the equivalent noise ˜
W=𝑃(ˆ
H𝑏)/𝑀 E𝑓X+Wis
not Gaussian distributed. However, since ˜
Whas uncorrelated
entries and is uncorrelated to 𝑃(ˆ
H𝑏)/𝑀 ˆ
H𝑓X,ithasthe
same first- and second-order moments as an additive white
Gaussian noise with variance ˜
𝜎2=𝜎2+𝑃(ˆ
H𝑏)∥E𝑓∥2
𝐹/𝑀 .
Therefore, if we replace ˜
Wwith a Gaussian noise with the
same variance, the resulting mutual information will be a
lower bound of the exact one [13]. Therefore, the outage
probability of the considered MIMO channel with transmit
power 𝑃(ˆ
H𝑏)and target rate 𝑅(ˆ
H𝑏)can be upper bounded by
𝒫𝑜𝑢𝑡 .
=𝒫
log det
I𝑁+𝑃(ˆ
H𝑏)ˆ
H𝑓ˆ
H†
𝑓
𝑀𝜎2+𝑃(ˆ
H𝑏)∥E𝑓∥2
𝐹
<𝑅(ˆ
H𝑏)
.
(5)
By letting e=[𝑒1,𝑒
2, ..., 𝑒𝑁],0<𝑒
1≤𝑒2≤... ≤𝑒𝑁,
denote the eigenvalue vector of E𝑓E†
𝑓, the joint probability
density function (pdf) of ecan be shown to be [15]
𝑝(e)=𝜉(𝜎2
𝑒,𝑓 )−𝑀𝑁
𝑁
𝑛=1
𝑒𝑀−𝑁
𝑛
𝑛<𝑘
(𝑒𝑛−𝑒𝑘)2exp−𝑁
𝑛=1𝑒𝑛
𝜎2
𝑒,𝑓
(6)
where 𝜉is a normalizing constant. Let 𝑝denote the exponential
order of 𝑃(ˆ
H𝑏). Thus, 𝜎2
𝑒,𝑓 ∝𝑃(ˆ
H𝑏)−1.
=𝜌−𝑝. Letting 𝑧𝑛
denote the exponential order of 1/𝑒𝑛, i.e., 𝑒𝑛.
=𝜌−𝑧𝑛,the
joint pdf of vector z=[𝑧1, ..., 𝑧𝑁]can be shown to be
𝑝(z).
=0,for any 𝑧𝑛<𝑝
𝑁
𝑛=1 𝜌−(2𝑛−1+𝑀−𝑁)(𝑧𝑛−𝑝),for all 𝑧𝑛≥𝑝.(7)
The outage probability in (5) can thus be shown to be
exponentially equal to
𝒫
log det
I𝑁+𝑃(ˆ
H𝑏)ˆ
H𝑓ˆ
H†
𝑓
𝑀𝜎2+𝜌𝑝𝑁
𝑛=1 𝜌−𝑧𝑛
<𝑅(ˆ
H𝑏)
.
=𝒫log det I𝑁+𝑃(ˆ
H𝑏)ˆ
H𝑓ˆ
H†
𝑓<𝑅(ˆ
H𝑏).(8)
With noisy CSITR, we propose two rate control schemes as
in Propositions 1 and 2 to mitigate the channel uncertainty and
minimize the outage probability. For notational convenience,
we let b=[𝑏1, ..., 𝑏𝑁],0<𝑏
1≤... ≤𝑏𝑁, denote the
eigenvalue vector of ˆ
H𝑏ˆ
H†
𝑏.
Proposition 1: Given ˆ
H𝑏, the transmitter adaptively adjusts
its data rate 𝑅(ˆ
H𝑏)in a way that
𝑟(ˆ
H𝑏)=¯𝑟+ lim
𝜌→∞
𝜏log det( ˆ
H𝑏ˆ
H†
𝑏)
log(𝜌)+
(9)
where 𝜏is a finite constant satisfying 𝜏≥max(1,1/𝛼).
70 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 1, JANUARY 2011
Now we prove the above rate control scheme satisfies the
average multiplexing gain constraint 𝐸{𝑟(ˆ
H𝑏)}=¯𝑟.
Proof: To get 𝐸{𝑟(ˆ
H𝑏)}, we only need to integrate over
the range where det( ˆ
H𝑏ˆ
H†
𝑏)=𝑁
𝑛=1 𝑏𝑛≤1,sincefor
any 𝑏𝑛>1, the joint pdf of 𝑏1, ..., 𝑏𝑁decays with SNR
exponentially [1]. Therefore,
𝐸¯𝑟+ lim
𝜌→∞
𝜏log 1
log(𝜌)≥𝐸{𝑟(ˆ
H𝑏)}
≥𝐸¯𝑟+ lim
𝜌→∞
𝜏log det( ˆ
H𝑏ˆ
H†
𝑏)
log(𝜌).
It follows from [13] that
lim
𝜌→∞
𝐸log det( ˆ
H𝑏ˆ
H†
𝑏)
log(𝜌)
= lim
𝜌→∞ 𝑁
𝑖=𝑀−𝑁+1 𝒳2
2𝑖
log(𝜌)=0
where 𝒳2
2𝑖is a chi-square random variable of dimension 2𝑖.
Since 𝜏is a finite constant, we have
𝐸¯𝑟+ lim
𝜌→∞
𝜏log 1
log(𝜌)=𝐸¯𝑟+ lim
𝜌→∞
𝜏log det( ˆ
H𝑏ˆ
H†
𝑏)
log(𝜌)=¯𝑟
which directly leads to 𝐸{𝑟(ˆ
H𝑏)}=¯𝑟.For𝜏to be a finite
constant, i.e., not to scale with log(𝜌), we require 𝛼∕=0.In
fact, it can be seen later that the value of 𝜏does not have
any impact on the D-M tradeoff, as long as 𝜏is a constant
satisfying 𝜏≥max(1,1/𝛼).
For some applications, a minimum data rate 𝑅𝑚𝑖𝑛 is
required to meet an acceptable quality of service [14]. Without
𝑅𝑚𝑖𝑛, it is also not very meaningful to discuss outage, because
the transmitter may switch off transmission completely if the
channel is too bad. To express 𝑅𝑚𝑖𝑛 in the limit of high
SNR, we adopt the concept of the minimum multiplexing
gain, which is defined as 𝑟𝑚𝑖𝑛 ≜lim𝜌→∞ 𝑅𝑚𝑖𝑛/log(𝜌)[6].
In Proposition 2, we propose a rate control scheme 𝑅(ˆ
H𝑏)in
thepresenceof𝑟𝑚𝑖𝑛. Analogously it can also be proved that
𝐸{𝑟(ˆ
H𝑏)}=¯𝑟.
Proposition 2: Given ˆ
H𝑏and 𝑟𝑚𝑖𝑛, the transmitter adjusts
its data rate 𝑅(ˆ
H𝑏)in a way that
𝑟(ˆ
H𝑏)=max¯𝑟+ lim
𝜌→∞
𝜏log det( ˆ
H𝑏ˆ
H†
𝑏)
log(𝜌),𝑟
𝑚𝑖𝑛.(10)
With the rate control schemes in Propositions 1 and 2, we
reach the following theorem.
Theorem 1: Consider an 𝑀×𝑁(𝑀≥𝑁) MIMO channel
with noisy CSITR obtained from the two-way training in
Section II. The achievable D-M tradeoff with average spatial
multiplexing gain ¯𝑟(¯𝑟≤𝑁) is characterized by
Case 1: If there is no minimum multiplexing gain constraint,
with the rate control scheme in Proposition 1, the achievable
diversity gain is infinity.
Case 2: In the presence of a nonzero 𝑟𝑚𝑖𝑛, with the rate
control scheme in Proposition 2, the achievable D-M tradeoff
is given by 𝑑(¯𝑟)=𝑑𝑛𝑜𝑃 𝐶 (𝑟𝑚𝑖𝑛)for any ¯𝑟∈[𝑟𝑚𝑖𝑛 ,𝑁].
Moreover, if the power control in [7] is performed together
with this rate control, the achievable D-M tradeoff is given
by 𝑑(¯𝑟)=𝑑𝑃𝐶(𝑟𝑚𝑖𝑛)for any ¯𝑟∈[𝑟𝑚𝑖𝑛,𝑁]. Here, 𝑑𝑃𝐶(𝑟)
denotes the achievable D-M tradeoff with power control based
on noisy CSIT, given in our earlier work [7].
It can be verified immediately that if ¯𝑟=𝑟𝑚𝑖𝑛 , we will
have 𝑟(ˆ
H𝑏)=𝑟𝑚𝑖𝑛, which corresponds to the case of no rate
control. In this case, the results in Theorem 1 reduce to those
in [1] and [7] without rate control.
IV. PROOF OF THEOREM 1
We let 𝑣𝑛,𝑤𝑛and 𝑢𝑛denote the the exponential orders of
1/𝑎𝑛,1/𝑏𝑛and 1/𝑑𝑛, respectively, where a=[𝑎1, ..., 𝑎𝑁],
0<𝑎
1≤... ≤𝑎𝑁,andd=[𝑑1, ..., 𝑑𝑁],0<𝑑
1≤... ≤𝑑𝑁,
denote the eigenvalue vectors of HH†and ˆ
H𝑓ˆ
H†
𝑓, respectively.
It can be easily shown that det( ˆ
H𝑏ˆ
H†
𝑏).
=𝜌−∑𝑛𝑤𝑛and
det(I𝑁+𝑃(ˆ
H𝑏)ˆ
H𝑓ˆ
H†
𝑓).
=𝜌∑𝑛(𝑝−𝑢𝑛)+.
To calculate the outage probability, we need to find out the
joint pdf of 𝑣𝑛,𝑤𝑛and 𝑢𝑛,𝑛=1, ..., 𝑁 . The joint pdf of
v=[𝑣1, ..., 𝑣𝑁]at high SNRs is given by [1]
𝑝(v).
=0,for any 𝑣𝑛<0
𝑁
𝑛=1 𝜌−(2𝑛−1+𝑀−𝑁)𝑣𝑛,for all 𝑣𝑛≥0.(11)
Given v, the conditional joint pdf of w=[𝑤1, ..., 𝑤𝑁]can
be shown to be
𝑝(w∣v).
=
𝑁1
𝑛=1
𝜌−(2𝑛−1+𝑀−𝑁)(𝑤𝑛−𝛼)(12)
if min(𝑣𝑛,𝑤
𝑛)≥𝛼for 𝑛=1, ..., 𝑁1,and0≤𝑣𝑛=𝑤𝑛<𝛼
for 𝑛=𝑁1+1, ..., 𝑁 ;otherwise𝑝(w∣v)=0.
Given vand w, the conditional joint pdf of u=[𝑢1, ..., 𝑢𝑁]
can be shown to be
𝑝(u∣w,v).
=
𝑁2
𝑛=1
𝜌−(2𝑛−1+𝑀−𝑁)(𝑢𝑛−𝑝)(13)
if min(𝑣𝑛,𝑢
𝑛)≥𝑝for 𝑛=1, ..., 𝑁2,and0≤𝑣𝑛=𝑢𝑛<𝑝
for 𝑛=𝑁2+1, ..., 𝑁 ;otherwise𝑝(u∣w,v)=0.Herewehave
𝑝=1with no power control and 𝑝=1+𝑁
𝑛=1(2𝑛−1+
𝑀−𝑁)𝑤𝑛≥1with the power control scheme in [7] applied.
By combining (11), (12) and (13), the joint pdf of v,uand w
is given by 𝑝(v,u,w)=𝑝(u∣w,v)𝑝(w∣v)𝑝(v).
A. Rate Control Without 𝑟𝑚𝑖𝑛 Constraint
With the rate control in Proposition 1, the outage probability
in (8) can be obtained as
𝒫𝑜𝑢𝑡 .
=𝒫𝑁
𝑛=1
(𝑝−𝑢𝑛)++𝜏
𝑁
𝑛=1
𝑤𝑛<¯𝑟
=∑𝑁
𝑛=1
(𝑝−𝑢𝑛)++𝜏∑𝑁
𝑛=1𝑤𝑛<¯𝑟
𝑝(v,u,w)𝑑v𝑑u𝑑w.(14)
For 𝑝(v,u,w)∕=0, we either have 𝑢𝑛=𝑣𝑛<𝑝or 𝑢𝑛,𝑣
𝑛≥
𝑝, which leads to (𝑝−𝑢𝑛)+=(𝑝−𝑣𝑛)+. Therefore, we may
simply replace 𝑢𝑛with 𝑣𝑛in (14). We examine the outage
event and consider the following two cases.
Case 1: If 𝑣𝑛≥𝛼,then𝑤𝑛≥𝛼and therefore we have
(𝑝−𝑣𝑛)++𝜏𝑤𝑛≥𝜏𝛼 ≥1.
Case 2: If 𝑣𝑛<𝛼,then𝑣𝑛=𝑤𝑛and therefore (𝑝−𝑣𝑛)++
𝜏𝑤𝑛=(𝑝−𝑣𝑛)++𝜏𝑣𝑛≥𝑝≥1.
ZHANG et al.: ON THE DIVERSITY GAIN IN MIMO CHANNELS WITH JOINT RATE AND POWER CONTROL BASED ON NOISY CSITR 71
As a result, we have 𝑁
𝑛=1(𝑝−𝑣𝑛)++𝜏𝑁
𝑛=1 𝑤𝑛≥𝑁,
implying that 𝒫𝑜𝑢𝑡 =0. It is thus possible to operate at an
average spatial multiplexing gain ¯𝑟∈[0,𝑁]reliably by per-
forming the rate control scheme in Proposition 1. One possible
scenario is that if the channel is too bad, the transmitter may
choose not to send anything (𝑅(ˆ
H𝑏)→0). In this way, no
outage will occur.
B. Rate Control With 𝑟𝑚𝑖𝑛 Constraint
In the presence of a nonzero minimum multiplexing gain
constraint 𝑟𝑚𝑖𝑛, the outage probability with the rate control
scheme in Proposition 2 can be obtained as
𝒫𝑜𝑢𝑡 .
=𝒫𝑁
𝑛=1
(𝑝−𝑢𝑛)+
<max ¯𝑟+ lim
𝜌→∞
𝜏log det( ˆ
H𝑏ˆ
H†
𝑏)
log(𝜌),𝑟
𝑚𝑖𝑛
(𝑎)
=𝒫𝑁
𝑛=1
(𝑝−𝑣𝑛)+<𝑟
𝑚𝑖𝑛(15)
where (𝑎)is due to the fact that (𝑝−𝑢𝑛)+=(𝑝−𝑣𝑛)+
and 𝒫𝑁
𝑛=1(𝑝−𝑢𝑛)++𝜏𝑁
𝑛=1𝑤𝑛<¯𝑟=0. As a result, the
outage probability in (15) can be rewritten as
𝒫𝑜𝑢𝑡 .
=𝜌−𝑑(¯𝑟)(16)
where
𝑑(¯𝑟)≜min
𝑤𝑛,𝑢𝑛,𝑣𝑛∈𝒪+
𝑁1
𝑛=1
(2𝑛−1+𝑀−𝑁)(𝑤𝑛−𝛼)+
𝑁2
𝑛=1
(2𝑛−1+𝑀−𝑁)(𝑢𝑛−𝑝)+
𝑁
𝑛=1
(2𝑛−1+𝑀−𝑁)𝑣𝑛(17)
with
𝒪≜𝑤𝑛,𝑢
𝑛,𝑣
𝑛
𝑁
𝑛=1
(𝑝−𝑣𝑛)+<𝑟
𝑚𝑖𝑛 ;0≤𝑁1,𝑁
2≤𝑁;
min(𝑣𝑛,𝑤
𝑛)≥𝛼, for 𝑛=1, ..., 𝑁1;
𝑣𝑛=𝑤𝑛<𝛼,for 𝑛=𝑁1+1, ..., 𝑁 ;
min(𝑣𝑛,𝑢
𝑛)≥𝑝, for 𝑛=1, ..., 𝑁2;
𝑣𝑛=𝑢𝑛<𝑝,for 𝑛=𝑁2+1, ..., 𝑁 .
Next we find the optimal solution of 𝑣𝑛,𝑤
𝑛,𝑢
𝑛that mini-
mizes the SNR exponent 𝒢≜𝑁1
𝑛=1(2𝑛−1+𝑀−𝑁)(𝑤𝑛−
𝛼)+𝑁2
𝑛=1(2𝑛−1+𝑀−𝑁)(𝑢𝑛−𝑝)+𝑁
𝑛=1(2𝑛−1+
𝑀−𝑁)𝑣𝑛under 𝒪+. Noting that 𝒢is an increasing function
of 𝑢1, ..., 𝑢𝑁2, and the minimizing solution of 𝑢1, ..., 𝑢𝑁2
should be 𝑢∗
1=... =𝑢∗
𝑁2=𝑝. Substituting the solution
into 𝒢,weget𝒢=𝑁1
𝑛=1(2𝑛−1+𝑀−𝑁)(𝑤𝑛−𝛼)+
𝑁
𝑛=1(2𝑛−1+𝑀−𝑁)𝑣𝑛, which is now an increasing
function of 𝑤1, ..., 𝑤𝑁1. Combining with the fact that decreas-
ing 𝑤1, ..., 𝑤𝑁1does not violate the outage condition in 𝒪+,
i.e., 𝑁
𝑛=1(𝑝−𝑣𝑛)+<𝑟
𝑚𝑖𝑛, the minimizing solution of
𝑤1, ..., 𝑤𝑁1should be 𝑤∗
1=... =𝑤∗
𝑁1=𝛼. Substituting
these solutions into (17), it follows that
𝑑(¯𝑟)= min
𝑣𝑛∈˜
𝒪+
𝑁
𝑛=1
(2𝑛−1+𝑀−𝑁)𝑣𝑛(18)
0 0.2 0.4 0.6 0.8 1
−50
0
50
100
150
200
250
CSIT quality α
Diversity gain
Joint rate and power control with imperfect CSITR
Power control with imperfect CSIT and perfect CSIR [7]
Rate control with imperfect CSITR
No CSIT, perfect CSIR [1]
Fig. 1. Diversity versus CSIT quality in a 5×3MIMO channel with
𝑟𝑚𝑖𝑛 =1.𝑟or ¯𝑟is chosen to be 3 in this figure.
where ˜
𝒪≜𝑣𝑛∣𝑁
𝑛=1(𝑝−𝑣𝑛)+<𝑟
𝑚𝑖𝑛with 𝑝being
equal to 1+𝑁
𝑛=1(2𝑛−1+𝑀−𝑁) min(𝛼, 𝑣𝑛)when there
is power control and being equal to 1when there is no power
control. It turns out that the original optimization problem
in (17) has been transformed into an optimization problem
solved in literature. If the transmitter does not perform power
control, the optimization problem in (18) is equivalent to the
problem solved in [1] for the MIMO channel with no CSIT
and perfect CSIR at spatial multiplexing gain 𝑟=𝑟𝑚𝑖𝑛.That
is, 𝑑(¯𝑟)=𝑑𝑛𝑜𝑃 𝐶 (𝑟𝑚𝑖𝑛). On the other hand, if power control
is performed at the transmitter, the optimization problem in
(18) is equivalent to the problem solved in [7] for the MIMO
channel with imperfect CSIT and perfect CSIR at spatial
multiplexing gain 𝑟=𝑟𝑚𝑖𝑛.Thatis,𝑑(¯𝑟)=𝑑𝑃𝐶(𝑟𝑚𝑖𝑛 ).The
proof of Theorem 1 is thus completed.
V. D ISCUSSIONS AND CONCLUSIONS
As shown in Theorem 1, if there is no minimum multiplex-
ing gain constraint 𝑟𝑚𝑖𝑛, rate control itself is able to boost the
diversity gain unboundedly, since the transmitter may choose
sufficiently low data rate to accommodate very bad channels.
It is in sharp contrast to the power control scheme which only
yields a finite diversity improvement as shown in [7]. On the
other hand, if there exists a nonzero minimum multiplexing
gain constraint 𝑟𝑚𝑖𝑛, outage events cannot be avoided since
there is a finite probability that the channel cannot support
𝑟𝑚𝑖𝑛. Therefore, rate control itself leads to a finite diversity
increase and the diversity gain is limited by 𝑑𝑛𝑜𝑃 𝐶 (𝑟𝑚𝑖𝑛).
In Fig. 1, we show the achievable diversity gain versus the
CSIT quality in a 5×3channel. It is interesting to find that
the actual value of 𝛼does not affect the achievable diversity
gain, if rate control is performed only. The reason is that with
rate control, the target multiplexing gain is determined by the
maximum value between 𝑟𝑚𝑖𝑛 and the rate specified in (9).
We have proved in Section IV.A that the system is always able
to support the rate in (9). Therefore, the outage occurs only
when 𝑟𝑚𝑖𝑛 is greater than the rate in (9) and the system cannot
support 𝑟𝑚𝑖𝑛.Since𝑟𝑚𝑖𝑛 is not dependent on 𝛼,thevalue
of 𝛼does not have any impact on the achievable diversity
72 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 1, JANUARY 2011
0.5 1 1.5 2 2.5 3
0
5
10
15
20
25
30
35
40
45
50
55
Multiplexing gain r
Diversity gain
Joint rate and power control with imperfect CSITR
Power control with imperfect CSIT and perfect CSIR [7]
Rate control with imperfect CSITR
No CSIT, perfect CSIR [1]
α=1/2
α=1/3
Fig. 2. D-M tradeoff in a 3×3MIMO channel with 𝑟𝑚𝑖𝑛 =0.5.The
multiplexing gain 𝑟on x-axis refers to ¯𝑟when rate control is involved.
gain. Clearly, this is different from power control, where the
achievable diversity gain highly depends on the value of 𝛼.
Fig. 2 shows the achievable D-M tradeoff in a 3×3
channel. Again, we can see that rate control based on noisy
CSITR leads to a diversity gain improvement over the diversity
gain with no CSIT and perfect CSIR [1]. With joint rate
and power control, the achievable diversity gain is further
improved dramatically and can be retained at the diversity gain
for 𝑟𝑚𝑖𝑛 in the entire range of average spatial multiplexing
gains, i.e., 𝑟𝑚𝑖𝑛 ≤¯𝑟≤𝑁.BothFigs.1and2alsoshow
that when 𝛼<1/(𝑀−1), power control itself brings only
marginal improvement over the case with no CSIT and perfect
CSIR, especially at high multiplexing gains. Together with rate
control, the diversity gain at any possible ¯𝑟can be boosted
to 𝑑𝑃𝐶(𝑟𝑚𝑖𝑛). This clearly shows that joint rate and power
control significantly improves the achievable D-M tradeoff
in MIMO channels with noisy CSITR, especially when the
multiplexing gain is high and the CSIT quality is poor.
We conclude this paper by comparing with the D-M tradeoff
for MIMO ARQ channels [16]. In ARQ channels, the receiver
feeds back to the transmitter a one-bit success/failure indicator.
By this means, the transmission rate is adaptively adjusted ac-
cording to the decoding status and the minimum multiplexing
gain is actually ¯𝑟/𝐿,where𝐿is the maximum number of
transmission rounds. If there is no delay constraint in ARQ
channels, i.e., 𝐿→∞, the outage can be completely avoided,
since the transmitter can always keep transmitting the same
message until the accumulated mutual information exceeds
the target data rate. This coincides with Case 1 of Theorem 1,
where there is no minimum multiplexing gain constraint. On
the other hand, if the ARQ protocol is allowed to use a finite
number of rounds, i.e., 𝐿<∞, the achievable diversity gain
(without power control) is given by 𝑑𝐴𝑅𝑄 =𝑑𝑛𝑜𝑃 𝐶 (¯𝑟/𝐿)
[16]. This coincides with Case 2 in Theorem 1 by letting
𝑟𝑚𝑖𝑛 =¯𝑟/𝐿. However, it should be noted that the ARQ
protocol assumes error-free backward links [16], while we
consider noisy backward links. From the D-M tradeoff point
of view, our rate control scheme with noisy feedback is able
to achieve the same performance as the ARQ protocol with
error-free feedback and is thus more advantageous. The work
presented in this paper has assumed MIMO Rayleigh fading
channels. As for the future work, it will be interesting to
extend the approach developed in this paper to multi-antenna
cooperative relaying channels [17], as well as non-Rayleigh
fading MIMO channels [18], [19].
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