Rate Gap Analysis for Rate-adaptive Antenna
Selection and Beamforming Schemes
Karthikeyan Shanmugam and Srikrishna Bhashyam
Department of Electrical Engineering
Indian Institute of Technology Madras
Chennai 600036, India
E-mail:
{karthikeyanshanmugam88@gmail.com, skrishna@ee.iitm.ac.in}
Abstract—We analyze the asymptotic performance of rate
adaptation for Transmit Antenna Selection (TAS) and Max-
imum Eigenmode Beamforming (MEB) schemes in Multiple-
Input Multiple-Output (MIMO) systems under imperfect channel
state information (CSI) and feedback delay. The rate is adapted
according to a target outage probability. We derive lower and
upper bounds to this rate. We also asymptotically characterize
the multi-step prediction error when MMSE prediction is used
to combat feedback delay. Using the bounds and the prediction
error asymptotics, we show that the rate gap from the ideal
CSI scenario asymptotically grows logarithmically with SNR. The
slope is at most the target outage probability. We find that when
the target outage probability is decreased faster than an identified
growth rate and prediction error goes to zero, then the rate gap
remains bounded.
I. INTRODUCTION
Several adaptive transmission schemes based on channel
state information (CSI) have been proposed for Multiple-
Input Multiple-Output (MIMO) wireless systems. Two of those
schemes are Maximum Eigenmode Beamforming (MEB) [1]
and Transmit antenna selection (TAS) [2]. The MEB scheme
involves beamforming along the eigen vector corresponding
to the largest singular value of the channel matrix. The
implementation of the MEB scheme requires feedback of at
least the beamforming vector assuming all other computations
are done at the receiver. The TAS scheme involves selecting
the best transmit antenna in terms of the maximum channel
norm. The TAS scheme has reduced complexity and requires
feedback of only the index of the transmit antenna to be
chosen. Also, the TAS scheme has been shown to achieve
full diversity asymptotically [3].
Rate or power adaptation can be employed along with the
above two schemes further to enhance performance [4]. We an-
alyze the performance of rate-adaptive MEB and TAS systems
in the presence of imperfections in CSI. When imperfections
in CSI at the receiver (due to estimation errors) and feedback
delay are introduced, even rate adaptation cannot always result
in an outage free transmission due to the mismatch between
estimates at the transmitter and the receiver. In other words,
the transmitter gets delayed information about changes in the
channel while receiver has information about both the current
and the past channel conditions. Transmitter adaptation has to
take place, under this uncertainty about CSI, at the transmitter.
One way to combat this problem is to use prediction. Proba-
bility of outage given a fixed rate at the transmitter has been
analysed for the MEB scheme in [5] for various imperfect CSI
assumptions. The average rate of a rate-adaptive TAS scheme
based on a fixed target outage probability has been numerically
calculated in [6].
In this paper, we derive analytical results for rate-adaptive
MEB and TAS schemes. First, we formally define the rate
adaptation scheme applied to the MEB and TAS systems. We
unify notation for the MEB and TAS schemes and define the
ergodic rate gap to be the expected difference between the rate
with perfect CSI and rate with imperfect CSI and feedback
delay. We derive lower and upper bounds to the adapted
rate with imperfect CSIT and show that the rate gap has a
log(SNR) growth. The slope of the rate gap is upper bounded
by the target outage probability. From this, we conclude that
when (1) the outage probability is decreased with SNR at a
rate faster than
1
log(SNR)
, and (2) channel prediction drives
the mismatch between CSI at the receiver (CSIR) and CSI at
the transmitter (CSIT) to zero when SNR becomes high, the
rate gap remains bounded.
As part of the above analysis, we also quantify the asymp-
totics of the mismatch (prediction error from noisy past
estimates) between CSIR and CSIT under multi-step MMSE
prediction for the Jakes fading model for the channel. The
asymptotics for one-step prediction with past values in noise
have been characterised in [7], [8] for Doppler processes.
We extend, by analytical calculations, the result to multi-step
prediction. We observe that the exact asymptotic variation of
prediction error with SNR does not have any implications for
the rate gap asymptotics as long as the prediction error goes
to zero.
The organisation of this paper is as follows. We present the
system model first in Section II, followed by rate adaptation
for both schemes in Section III. Then, we derive bounds on
the adapted rate for both the schemes in Section IV. We
characterize the asymptotics of prediction error with SNR
in Section V. In Section VI, we analyze the ergodic rate
gap, characterise it asymptotically, and present some numerical
results. Conclusions are drawn in Section VII.
978-1-4244-5637-6/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
II. SYSTEM MODEL
A MIMO system with N
t
transmit antennas and N
r
receive
antennas is considered. We assume a block rayleigh fading
channel. The correlation between different blocks follows the
Jakes fading model. The channel matrix for a particular block
is denoted by H and has i.i.d entries distributed as CN(0, 1).
The received vector y (N
r
× 1) is given by:
y =
√
P Hx + z (1)
where x is the transmitted signal vector and z ∼CN(0,σ
2
n
I).
The power used per training symbol is P
t
and the power used
per data symbol is P
d
. The estimated channel at receiver is
denoted H
r
. H
ij
r
∼CN(0,σ
2
r
). The CSIT is denoted H
t
and
H
ij
t
∼CN(0,σ
2
t
). We distinguish between two cases:
1) Perfect CSIR and no feedback delay: Here H
r
= H
t
and σ
2
r
= σ
2
t
=1.
2) Imperfect CSIR and feedback delay: The feedback delay
is Δ blocks. H
t
is obtained using a Δ-step channel prediction
from past values of H
r
.Letρ be the entry wise correlation
between H
r
and H
t
. σ
2
r
=
P
t
P
t
+σ
2
n
and σ
2
t
= p
H
w, where
w is the L-tap Wiener filter used. p is the cross-correlation
vector between the current H
ij
r
and past values with delay Δ.
The following relation holds (H
t
and H
r
are jointly Gaussian)
[5]:
H
r
= σ
r
ρ
σ
t
H
t
+
1 − ρ
2
E
(2)
where E
ij
∼CN(0, 1).
In the MEB scheme, the beamforming vector which cor-
responds to the largest singular value of the channel matrix
(CSIT) H
t
is selected. Let this beamforming vector be u.
Then, the transmit vector x = ux, where x is the transmitted
data symbol. In the TAS scheme, only one antenna is selected
for transmission. Therefore, x has only one non-zero entry
corresponding to the selected antenna.
III. R
ATE ADAPTAT ION
Rate adaptation for the TAS scheme was considered in [6].
Similarly, we consider rate adaptation for MEB scheme and
unify the rate gap analysis for both schemes. The transmission
rate is chosen based on a lower bound on the mutual infor-
mation and a target outage probability. For the MEB scheme,
the mutual information achievable at the receiver can be lower
bounded by [9],[10]:
I(x, y/H
t
, H
r
) ≥ log(1 + Γu
H
H
r
H
H
r
u) (3)
where Γ=
P
d
P
d
σ
2
e
+σ
2
n
. For a given H
t
, let the rate to be chosen
by the transmitter be R(H
t
). The probability of outage for this
rate can be upper bounded as in [6]:
P (outage/H
t
) ≤ P
A<2β
1+μ
σ
2
r
, (4)
where A =
2μ
σ
2
t
H
t
u +
√
2Eu
2
, μ =
ρ
2
1−ρ
2
, and β =
e
R(H
t
)
Γ
. I n order to ensure an upper bound on the outage
probability, the rate R(H
t
) is decided by equating t he upper
bound to a fixed outage probability P
out
and is given by:
R
0
(H
t
)=log
1+Γ
σ
2
r
2(1 + μ)
F
−1
nc−χ
2
,2N
r
,δ
(P
out
)
, (5)
where F
−1
nc−χ
2
,2N
r
,δ
is the inverse CDF of the non-central
χ
2
distribution with 2N
r
degrees of freedom and centrality
parameter δ =
2μ
σ
2
t
H
t
u
2
.Theβ corresponding to the above
R
0
is denoted β
0
.
The outage probability bound for the TAS scheme is also
very similar to the bound in equation (4). The only difference
is in the expression for A. H
t
u gets replaced by H
tsel
corresponding to the maximum norm column of H
t
at the
transmitter and Eu gets replaced by E
sel
. The statistics
of Eu and E
sel
are identical. Both are N
r
× 1 circularly
symmetric complex gaussian with variance
1
2
per dimension.
Therefore, conditioned on the CSIT (H
t
), the distribution
of random variable A is identical for both MEB and TAS
schemes. Let
ˆ
H denote H
tsel
and H
t
u in their respective
cases. Let
ˆ
E denote E
sel
and Eu in their respective schemes.
The distribution of A conditioned on H
t
is a non-central
chi-squared distribution with 2N
r
degrees of freedom and
centrality parameter δ =
2μ
σ
2
t
ˆ
H
2
.
Since the mutual information lower bound (outage upper
bound) is used for calculating the rate, we have
P
out
>P(outage/H
t
). (6)
Let the perfect CSI (H
t
= H
r
= H) rate be denoted by
R
ideal
. For the MEB scheme, the perfect CSI rate is:
R
ideal
= log(1 + SNR
˘
H
2
), (7)
where SNR = P
d
/σ
2
n
, and
˘
H = Hu, u is the singular vector
corresponding to the maximum singular-value of H, the actual
channel matrix. Since
ˆ
H
σ
t
in the imperfect CSI case and
˘
H in
the perfect CSI case have the same PDF, we first define ΔR
for a given
¯
H as follows:
ΔR(
¯
H)=R
ideal
(
¯
H)−(1−P (outage/
¯
Hσ
t
))R
0
(
¯
Hσ
t
), (8)
where R
ideal
(
¯
H) is the perfect CSI rate when
˘
H =
¯
H,
R
0
(
¯
Hσ
t
) is the imperfect CSI rate when
ˆ
H
σ
t
=
¯
H, and (1 −
P (outage/
¯
Hσ
t
) accounts for the possibility of outage with
imperfect CSIT. The ergodic rate gap will be E
¯
H
[ΔR(
¯
H)],
where
¯
H has the same PDF as
˘
H and
ˆ
H
σ
t
above.
Similarly, for the TAS scheme, the perfect CSI rate is
again given by (7). However, here
˘
H
2
is the maximum
channel norm over all transmit antennas. Again, considering
the imperfect CSI case in this scheme, the variable
ˆ
H
σ
t
2
is
statistically same as
˘
H
2
. Hence, ΔR(
¯
H) is given by the
same expression (8).
Although the rate gap expression is identical for both
schemes, the variables involved {
˘
H,
ˆ
H/σ
t
} are different statis-
tically. The TAS scheme is characterized by the statistics of the
maximum channel norm. The MEB scheme is characterized
by the statistics of the gaussian channel matrix multiplied by
the singular vector corresponding to the largest s ingular value.
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Hence, when ergodic rate gap is computed, the rate gap will
be averaged by different probability distributions.
The main result of the paper is to show the following
asymptotics for both TAS and MEB schemes:
E
¯
H
[ΔR(
¯
H)] <P
out
log(SNR)+O(1). (9)
IV. B
OUNDS ON THE RATE
To compute ΔR(
¯
H), we need to characterise R
0
(
¯
Hσ
t
)
or equivalently R
0
(
ˆ
H). The inverse CDF function in (5) is
numerically computable but difficult to characterise analyti-
cally. Therefore, to analytically characterize the asymptotics,
we bound the right-hand side of (4) in the following lemma.
Lemma 1. The conditional rate R
0
(
ˆ
H) has the following
bounds:
R
0
< log
1+Γρ
2
σ
2
r
σ
2
t
ˆ
H
2
+Γ
(1 − ρ
2
)σ
2
r
2
F
−1
χ
2
(2P
out
)
(10)
and
R
0
> log
⎛
⎝
1+Γσ
2
r
1 − ρ
2
2
F
χ
−1
(P
out
) −
ρ
σ
t
ˆ
H
2
⎞
⎠
(11)
where F
−1
χ
2
is the inverse CDF of the central χ
2
distribution
with 2N
r
degrees of freedom, and F
−1
χ
is the inverse CDF of
the chi distribution with 2N
r
degrees of freedom.
Proof: Consider the event E = {2
ˆ
E
2
< 2β
0
1+μ
σ
2
r
−
2μ
σ
2
t
ˆ
H
2
} . Since
ˆ
E is angularly symmetric (direction wise),
P (E
+
= {Re(
ˆ
H
H
ˆ
E) > 0})=P (E
−
= {Re(
ˆ
H
H
ˆ
E) < 0)}.
Also, we have
P (E
−
∩E) <P
A<2β
0
1+μ
σ
2
r
,
and
P (E
−
∩E)=P (E
+
∩E).
Therefore, using the above three equations, we get
1
2
P (E) <P
A<2β
0
1+μ
σ
2
r
If the lower bound on left hand side of the previous equation
is equated to the target outage probability P
out
, then a rate
greater than R
0
will be obtained. Therefore, we get the upper
bound as given in the lemma. The inverse CDF F
−1
χ
2
is due
to the statistics of 2
ˆ
E
2
.
In order to derived the rate lower bound, consider the
following triangle inequality:
√
2
ˆ
E +
2μ
σ
2
t
ˆ
H
2
≥
√
2
ˆ
E−
2μ
σ
2
r
ˆ
H
2
. (12)
Let P
A<
2β
0
(1+μ)
σ
2
r
be denoted by P
A
. The following
upper bound holds:
P
A
<P
⎛
⎝
√
2
ˆ
E−
2μ
σ
2
r
ˆ
H
2
< 2β
0
1+μ
σ
2
r
⎞
⎠
= P
√
2
ˆ
E−
2μ
σ
2
r
ˆ
H
≤
2β
0
1+μ
σ
2
r
≤ P
√
2
ˆ
E≤
2β
0
1+μ
σ
2
r
+
2μ
σ
2
t
ˆ
H
.
Equating the right hand side of the above inequality to P
out
,
we get the lower bound on the rate. F
−1
χ
is due to
√
2
ˆ
E.
We denote the upper bound by R
upp
(
ˆ
H) and lower bound
by R
low
(
ˆ
H) from now on.
V. A
SYMPTOTIC MISMATCH BETWEEN CSIR AND CSIT
In order to charaterize the asymptotic behavior of R
upp
and
R
low
, the asymptotics of 1 − ρ
2
needs to be characterised in
the imperfect CSI case. In this section, we show the following
assuming prediction using the entire past:
1 − ρ
2
= O(ln(SNR)
2(Δ−1)
SNR
−(1−
ω
m
π
)
). (13)
Each H
ij
t
[n] is predicted based on past CSIR {H
ij
r
[n −
Δ],H
ij
r
[n −Δ −1]...., H
ij
r
[n −Δ −L +1]}. Note that H
ij
r
is
the MMSE estimate of H
ij
. We assume that H
ij
is a Doppler
process with spectrum F (e
jω
). H
ij
r
is nothing but a Doppler
process H
ij
in noise. For the Doppler process in noise, the
power spectral density is given by:
S(e
jω
)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
P
2
t
F (e
jw
)
(P
t
+ σ
2
n
)
2
+
P
t
σ
2
n
(P
t
+ σ
2
n
)
2
|ω| <ω
m
P
t
σ
2
n
(P
t
+ σ
2
n
)
2
ω
m
< |ω| <π
,
(14)
Specifically, for the Jakes correlation model F (e
jω
) has the
following form:
F (e
jω
)=
2
ω
m
1 −
ω
ω
m
2
(15)
for |ω| <ω
m
. Since MMSE prediction is used, we have
σ
2
t
+ σ
2
p
= σ
2
r
, where σ
2
p
is the prediction error and ρ =
σ
t
σ
r
.
Therefore, 1 − ρ
2
=
σ
2
p
σ
2
r
. Since,
1
σ
2
r
= O(1) at high SNR,
1 −ρ
2
is dependent on the prediction error. This characterizes
the mismatch between CSIR and CSIT. The noise power in
the case where H
ij
r
is normalised with σ
r
is
1
1+SNR
.
As noted before, for Doppler processes, asymptotics of one
step (Δ=1) prediction error in noise (with power
1
SNR
)
has been characterised in [7], [8]. This analysis is the best
case scenario when the entire past is used in prediction (holds
almost for large L). We quote the result here:
1 − ρ
2
= σ
2
p
= O(SNR
−
(
1−
ω
m
π
)
). (16)
978-1-4244-5637-6/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
We now derive a similar result for multi-step prediction using
results from [11]. The correlation model assumed in (14) obeys
Paley-Wiener condition (this can occur when F is absolutely
log integrable in its support) which is:
π
−π
log(S(e
jω
))dω > −∞
It can be noted that the Jakes spectrum in noise also satisfies
this criterion. The following hold for S(e
jω
)
ln(S(e
jω
)) =
n=∞
n=−∞
c
n
e
−jωn
and, therefore, we have
S(e
jω
)=e
c
0
2
+
∞
n=1
c
n
e
−jωn
e
c
0
2
+
−∞
n=−1
c
n
e
−jωn
= S
+
(e
jω
)S
−
(e
jω
).
Let f
n
be defined such that:
S
+
(e
jω
)=
∞
n=0
f
n
e
−jωn
.
Expanding S
+
to get f
n
,wehave:
S
+
(e
jω
)=e
c
0
2
⎡
⎢
⎢
⎢
⎣
∞
k=0
∞
n=1
c
n
e
−jωn
k
k!
⎤
⎥
⎥
⎥
⎦
(17)
= e
c
0
2
1+c
1
e
−jω
+
c
2
+
c
2
1
2!
e
−2jω
+ ..
n
k=1
i
1
+..i
k
=n
(c
i
1
..c
i
k
)
k!
e
−jωn
+ ..
.
Now, c
n
is evaluated as follows:
c
n
=
1
2π
⎛
⎝
ω
m
−ω
m
ln(
2
+ F (e
jωn
))e
jωn
dω+
−ω
m
−π
ln(
2
)e
jωn
dω +
π
ω
m
ln(
2
)e
jωn
dω
⎞
⎠
= O(1) + O(ln(
2
))
= O(1) + O(ln(SNR)),
where
2
=
P
t
σ
2
n
(P
t
+σ
2
n
)
2
= O(SNR
−1
) is the noise variance. Let
E
N
denote the N- step prediction error. Then, we have
E
N
=
N−1
n=0
f
2
n
. (18)
Also, e
c
0
= O(SNR
−(1−
ω
m
π
)
). When f
n
is expressed in
terms of c
n
’s using equation (17), it is a summation of terms of
the form c
i
1
...c
i
k
(product of some c
n
’s). The dominant term in
f
2
n
, from terms like c
i
1
...c
i
k
,isO(ln(SNR)
2n
SNR
−(1−
ω
m
π
)
).
This is due to the term
c
n
1
n!
. Therefore, we have
1 − ρ
2
= σ
2
p
= E
Δ
= O(ln(SNR)
2(Δ−1)
SNR
−(1−
ω
m
π
)
)
(19)
Here Δ is the delay as mentioned before. When Δ=1,we
get back the existing result in (16).
VI. E
RGODIC RATE GAP ANA LYS IS
In this section, we characterise asymptotics of ΔR(
¯
H) and
show the result stated in (9). The following lemma holds.
Lemma 2. If R
ideal
(
¯
H) denotes the rate under perfect CSI,
and R
low
(
¯
Hσ
t
) and R
upp
(
¯
Hσ
t
) are the upper and lower
bounds on the rate for imperfect CSI R
0
(
¯
Hσ
t
) with delay
Δ, then
lim
SNR→∞
R
ideal
−R
low
= lim
SNR→∞
R
ideal
−R
upp
= log(1+
1
η
)
where P
d
= ηP
t
.
Proof: Let x = SNR
¯
H
2
, and y =Γρ
2
σ
2
r
¯
H
2
+
Γσ
2
r
1−ρ
2
2
F
−1
χ
2
(2P
out
). We observe that
lim
SNR→∞
R
ideal
−R
upp
= log
1+x
1+y
= log
1+
x
y
− 1
1+
1
y
.
(20)
Hence, it is enough to characterise the ratio
y
x
. Also, note that
Γρ
2
σ
2
r
=Γσ
2
r
− Γ(1 − ρ
2
)σ
2
r
,
lim
SNR→∞
Γσ
2
r
SNR
=
1
1+
1
η
, and
lim
SNR→∞
1 − ρ
2
=0.
The last equality follows from (19). Therefore, the following
holds:
lim
SNR→∞
y
x
=
1
1+
1
η
(21)
By eqns (21) and (20), we have
lim
SNR→∞
R
ideal
− R
upp
= log(1 +
1
η
)
Similarly, the lower bound result can also be proved by
using y =Γσ
2
r
1−ρ
2
2
F
χ
−1
(P
out
) − ρ
¯
H
2
.
In the above result, note that while the prediction error goes
to zero, the rate at which the prediction error tends to zero does
not matter. This is different from the diversity-multiplexing
gain tradeoff in [5] where the rate at which ρ tends to 1 is
important. The rate gap as defined in (8) can be bounded on
both sides using (6) and R
low
,R
upp
as:
R
ideal
(
¯
H) − (1 − P (outage/
¯
Hσ
t
))R
upp
(
¯
Hσ
t
) < ΔR(
¯
H)
<R
ideal
(
¯
H) − (1 − P
out
)R
low
(
¯
Hσ
t
)
⇒ P (outage/
¯
Hσ
t
)R
ideal
+(1− P (outage/
¯
Hσ
t
))(R
ideal
−
R
upp
) < ΔR<P
out
R
ideal
+(1− P
out
)(R
ideal
− R
low
).
The second term of each bound above goes to a con-
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5 10 15 20 25 30 35 40
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
SNR(in dB)
Ergodic rate diff in nats/sec/Hz
P
out
=0.05
P
out
=0.08
P
out
=0.01*log(10
0.8
)/log(SNR)
P
out
=0.01*log(10
0.8
)
2
/log(SNR)
2
P
out
=0.01*(10
0.8
)/SNR
Fig. 1. Ergodic rate gap for a 2 × 2 system with ω
m
=0.1π,N
r
= N
t
=2,
Δ=2and L =20under TAS scheme
stant by Lemma 2. Therefore, the growth rate of ΔR is
upper bounded by P
out
log(O(SNR)) and lower bounded by
P (outage/
¯
Hσ
t
) log(O(SNR)), i.e.,
ΔR(x) < K(x)P
out
log(SNR)
for SNR > Θ where x is a realisation of the variable
¯
H and
Θ is independent of the variable x (from the definition of O(.)
notation).
Now, the ergodic rate gap is upper bounded as follows:
∞
0
ΔR(x)f
¯
H
2
(x)dx <
∞
0
(P
out
log(SNR)+K(x))f
¯
H
2
(x)dx
(22)
for SNR > Θ where Θ is independent of the variable
x. f
¯
H
2
is the pdf according to the scheme chosen. On
averaging, the second term results in a constant. The first term
integrates out to P
out
log(SNR). Hence the result in (9) has
been shown.
From the above results, we observe that if P
out
(outage
target) is not a function of SNR, the ergodic rate gap
grows as log(SNR). However, if P
out
is decays faster than
(log(SNR))
−1
, then the rate gap is bounded. This means
that at high SNR, the outage target should be lower. Note
that Lemma 2 still holds since all inverse CDF functions also
decrease as P
out
decreases .
We also show the rate gap behaviour with respect to the
chosen outage probability through simulations under TAS
scheme (ref. Fig.1). The ergodic rate gap (actually with P
out
substituted in (8) and inverse CDFs of non-central distribu-
tion calculated numerically) is computed using monte carlo
simulations (to average different realisations
¯
H) and plotted
for various values of SNR. Δ=2, N
r
= N
t
=2and
ω
m
=0.1π are the parameters used. We observe that the rate
gap growth for very high SNR is linear in the constant P
out
case and the slope increases when P
out
=0.05 is increased to
P
out
=0.08. Also, we note that for the following growth rates
SNR
−1
, log(SNR)
−1
, log(SNR)
−2
, the rate gap is bounded
asymptotically as predicted by the theory.
VII. C
ONCLUSIONS
We have compared the achievable rates under delay and
imperfect CSI of the rate adaptive TAS and MEB schemes with
that of the perfect cases respectively. The analysis is common
and the ergodic rate gap in both cases are shown to have a
log(SNR) growth. The slope depends on the target outage
probability. The rate gap is shown to be bounded if the target
outage probability is reduced with SNR.
It is instructive to note that the analysis for uplink MAC
user selection is similar to the TAS scheme and the analysis
for downlink BC user selection is similar to that of the MEB
scheme (except that the beamforming vector is the unit channel
vector of the chosen user and the maximum norm would be
the choosing criteria). The analysis presented here holds good
when prediction error goes to zero and MMSE filter with large
number of taps almost ensures it. Furthermore, the outage
probability has to be decided based on SNR and if suitably
chosen can make rate gap bounded.
A
CKNOWLEDGEMENT
This work was supported in part by the Department of
Science and Technology, Govt. of India.
R
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
