3674 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008
Exact BER Analysis for M-QAM Modulation with
Transmit Beamforming under Channel Prediction Errors
Eduardo Martos-Naya, Jos´e F. Paris, Unai Fern´andez-Plazaola, and Andrea J. Goldsmith, Fellow, IEEE
Abstract—Significant throughput improvements can be ob-
tained in multiple-input multiple-output (MIMO) fading channels
by merging beamforming at the transmitter and maximal ratio
combining (MRC) at the receiver. In general, accurate channel
state information (CSI) is required to achieve these performance
gains. In this paper, we analyze the impact of channel prediction
error on the bit error rate (BER) of combined beamforming
and MRC in slow Rayleigh fading channels. Exact closed-form
BER expressions are obtained in terms of elementary functions.
Numerical results show that imperfect CSI causes little BER
degradation using channel prediction of moderate complexity.
Index Terms—Multiple input multiple output (MIMO) sys-
tems, beamforming, maximal ratio combining (MRC), imperfect
channel state information (ICSI), channel prediction, bit error
rate (BER).
I. INT RODUCTION
M
ULTIPLE-INPUT multiple-output (MIMO) systems
can considerably increase data rates through spatial
multiplexing and significantly improve robustness and cover-
age through beamforming and diversity combining [1]. The
capacity and performance of MIMO systems with m ulti-
plexing, beamforming and diversity depends on the avail-
ability and accuracy of the channel state information (CSI)
at both the transmitter and receiver. The impact of imp er-
fect CSI has been the subject of much recent investigation
(see e.g. the special issues [2] and [3]).
This paper focuses on the impact of imperfect CSI on
MIMO beamforming, which has been recently addressed in
[4], [5], [6], [7]. In particular, [7] determined the pdf, cdf,
and moment-generating function (MGF) of the output SNR
in transmit beamforming under imperfect CSI. The system
model assumed in [7] considers the same CSI to perform
both beamforming at the transmitter and MRC at the receiver.
However, in this paper we focus on a different system model
which considers a more accurate CSI for MRC than for
beamforming, as was also considered in [4], [5], [6]. These
papers investigated the impact of imperfect CSI on transmit
beamforming combined with MRC at the receiver and adaptive
modulation. These works do not attempt to obtain exact
closed-form BER expressions but rather use approximations,
typically based on exponential bounds (e.g. [8, eq. 17]).
Manuscript received February 15, 2007; re vised May 25, 2007, September
10, 2007, January 16, 2008, and May 30, 2008; accepted July 31, 2008. The
associate editor coordinating the review of this paper and approving it for
publication was I. Collings.
E. Martos-Naya, J. F. Paris, and U. Fern´andez-Plazola are with the
Departamento de Ingeniera de Comunicaciones, Universidad de M´alaga (e-
mail: eduardo@ic.uma.es).
A. J. Goldsmith is with the Department of Electrical Engineering, Stanford
Univ ersity.
Digital Object Identifier 10.1109/T-WC.2008.070192
In contrast, our analysis focuses on obtaining exact BER
expressions.
The BER for transmit beamforming will depend on the
effective channel gain. If the CSI available to perform beam-
forming at the transmitter is different from the CSI to perform
MRC at the receiver, the pdf of the effective channel gain
can seldom be obtained in closed-form. In fact, the well-
known pdf/MGF approach [9] cannot be extended to combined
beamforming and MRC analyzed in this paper, and thus
alternative analysis techniques are needed.
Under imperfect CSI caused by channel prediction errors,
we obtain exact closed-form BER expressions for transmit
beamforming using a different approach. Specifically, for a
system with fixed power and M-QAM constellation, we com-
pute the exact BER by first calculating the conditional BER,
conditioned on the predicted channel, using Proakis’ analysis
of complex Gaussian quadratic forms [10, Appendix B][11]
and then averaging the conditional BER over the distribution
of the predicted channel.
The remainder of this paper is organized as follows.
Section II describes the system model. In Section III the BER
expressions are derived. Section IV presents numerical results
which exploit the analytical expressions derived in previous
sections. Finally, conclusions are provided in Section V.
II. S
YSTEM MODEL
The system model for MIMO beamforming with MRC
is briefly described in this section. Further details on the
adopted system model can be found in [4]. We consider N
T
transmit antennas and N
R
receive antennas, and the channel
is modeled by an N
R
× N
T
complex matrix H, so that each
entry H
i,j
is the channel coefficient between the jth transmit
and the ith receive antennae. These channel coefficients exhibit
frequency-flat slowly time-varying fading. The entries H
i,j
are
assumed independent identically-distributed (i.i.d) complex
circularly symmetric normal random variables (RVs), with
zero-mean and unity-variance, i.e. H
i,j
∼CN(0, 1).Noise
is modeled by an additive N
R
-dimensional vector n, whose
entries n
k
are i.i.d. complex circularly symmetric normal RVs
∼CN(0,N
0
). The received signal can be expressed as
y = Hw + n, (1)
where y is the received N
R
dimensional complex vector and
w is the transmitted N
T
dimensional complex vector.
In our receiver model, as in [4], [5], [6], we assume an
imperfect channel prediction to obtain the predicted beam-
steering vector which mu st be fed back to the transmitter. The
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008 3675
predicted channel
ˆ
H can be expressed as follows
ˆ
H = H −
ˆ
Ξ , with
ˆ
H
i,j
i.i.d. RVs ∼CN(0, 1 − χ)
ˆ
Ξ
i,j
i.i.d. RVs ∼CN(0,χ)
,
(2)
where
ˆ
Ξ is the prediction error matrix and χ the mean square
error. We assume the entries of the predicted channel matrix
ˆ
H
i,j
and the entries of the prediction error matrix
ˆ
Ξ
i,j
are
orthogonal. As in [4], our analysis adopts a slowly time-
varying channel model in which the channel response remains
invariant along the frame in terval.
Using the predicted channel
ˆ
H, the optimal beam-steering
vector
ˆ
v is the N
T
-dimensional eigenvector corresponding
to the largest eigenvalue
ˆ
λ of matrix
ˆ
H
H
ˆ
H [12], which is
given by
ˆ
λ =
ˆ
v
H
ˆ
H
H
ˆ
H
ˆ
v. On each frame, the receiver feeds
vector
ˆ
v back to the transmitter to perform beamformin g,
so that the transmitted vector beco mes w = ˆvz,wherez
is the complex transmitted symbol. The effective channel
is an N
R
-dimensional vector defined as h
Δ
= Hˆv and the
predicted effective channel is the vector
ˆ
h
Δ
=
ˆ
Hˆv, whose
square Euclidean norm is ||
ˆ
h||
2
=
ˆ
λ. The effective channel can
also be expressed as h
Δ
= Hˆv =(
ˆ
H +
ˆ
Ξ)ˆv =
ˆ
h+Ψ,whereΨ
is a complex normal N
R
-dimension vector whose entries Ψ
k
,
assuming that
ˆ
v is a unitary vector, are i.i.d. RVs ∼CN(0,χ)
[13, p. 26].
At the receiver, the effective channel vector h = Hˆv is
assumed perfectly estimated to perform MRC. This assump-
tion is reasonable since the received signals can be stored
so that non-causal channel estimation (smoothing) with high
accuracy can be performed [14]. The symbol r which results
from applying MRC to received vector y is given by
r
Δ
=
h
H
y
||h||
2
= z + n
, with n
Δ
=
(
ˆ
h + Ψ)
H
n
||
ˆ
h + Ψ||
2
, (3)
where n
is the resultant noise after MRC. As the entries
of noise vector n are i.i.d circularly symmetric Gaussian
variables and h and n are mutually independent, it is straight-
forward to show that n
is circularly sy mmetric too.
III. BER A
NALY SI S
One approach to calculate the BER from (3) could be to
average the conditional BER over the effective SISO channel
gain ||h||
2
. Unfortunately, the pdf of the effective SISO
channel gain is unknown to the best of the authors’ knowledge.
For this reason we use an alternative analysis method. First,
we compute the BER conditioned on the predicted effective
channel vector
ˆ
h, using the Proakis’ analysis of Gaussian
quadratic forms [10, Appendix B][11]. As is shown in this
section, this conditional BER expression only depends on
ˆ
λ = ||
ˆ
h||
2
, whose pdf is directly related to the well-known
pdf of λ.
The BER analysis presented in this section considers
L-PAM or square M-QAM (M = L
2
) with independent bit-
mapping for in-phase and quadrature components, e.g, Gray
mapping. Under these assumptions and reminding that the
resultant noise n
is circularly symmetric, the BER can be
expressed, in a similar way to [15][16], as
BER =
L−1
n=1
ω(n)I(n), (4)
where I(n) are called components of error probabil-
ity (CEP) and the ω(n) are coefficients dependent on
the constellation mapping. The CEPs are defined as
I(n)=Pr{{n
} > (2n − 1)d},whered is the minimum
distance between the symbol and the decision boundary and
can be expressed as a function of the constellation energy E
s
as d =
6E
s
M−1
for M-QAM constellation and d =
12E
s
L
2
−1
for
L-PAM constellation. The coefficients ω(n) can be obtained
using the method described in [16] or directly computed for
Gray mapping as in [15].
In Section III-A we derive the CEP conditioned on the
predicted effective channel vector
ˆ
h. In Section III-B, using
the previous results, we obtain both the conditional BER and
the BER averaged over
ˆ
λ.
A. Conditional CEP
In this section we derive, for our system model, the CEP
conditioned on a predicted channel state (CCEP), that can be
expressed as
I(n;
ˆ
h)=Pr
{n
} > (2n − 1)d |
ˆ
h
. (5)
In order to obtain the CCEP we use Proakis’ analysis of
complex Gaussian quadratic forms [10, Appendix B], more
specifically, we adopt the compact expressions presented in
[11] and restated in Table I.
According to our system model, when the symbol z = s
is transmitted , each entry o f the received signal y
k
and each
entry o f the effective channel h
k
are
y
k
= h
k
s + n
k
=(
ˆ
h
k
+Ψ
k
)s + n
k
, (6)
h
k
=
ˆ
h
k
+Ψ
k
, (7)
where both y
k
and h
k
are complex normal variables condi-
tioned on the predicted effective
ˆ
h
k
. Note that
ˆ
h
k
is assumed
orthogonal to the channel error Ψ
k
(e.g.
ˆ
h
k
is predicted using
a FIR Wiener filter) and therefore, the probability distribution
of Ψ
k
is independent of the
ˆ
h
k
value. Defining the sum of
quadratic forms D =
N
R
k=1
x
H
k
Qx
k
,where
x
k
=
y
k
h
k
, Q =
0 −1/2
−1/2 {s} +(2n − 1)d
,
(8)
we have
Pr{D<0} =
Pr{−{h
H
y} +({s} +(2n − 1)d)||h||
2
< 0 |
ˆ
h} =
Pr{{n
} > (2n − 1)d |
ˆ
h} = I(n;
ˆ
h).
(9)
In Ta ble I we present the expressions to obtain this probability.
Using the quadratic form matrix Q, the mean vector m
k
and
the covariance matrix R of vector x
k
, we can calculate the
parameters a, b, η and C
l
necessary to obtain this probability.
From (6) and (7), we obtain the mean and the covariance
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3676 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008
TABLE I
P
ROBABILITY COMPUTAT ION O F T H E GAUSSI AN QUADRATI C FORM.
FUNCTIONS AND PARAMETERS DE FINITIONS
D
L
k=1
x
H
k
Qx
k
m
k
E {x
k
}
R E
(x
k
− m
k
)(x
H
k
− m
H
k
)
{δ
i
}
i=1,2
1
2
tr(RQ) ±
1
2
tr(RQ)
2
− det(RQ)
η
δ
1
δ
2
a
2δ
2
Σ
L
k=1
m
H
k
Q − δ
1
R
−1
m
k
(δ
1
− δ
2
)
2
b
2δ
1
Σ
L
k=1
m
H
k
Q − δ
2
R
−1
m
k
(δ
1
− δ
2
)
2
C
l
(a, b, η)
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
−1+
1
(1 + η)
2L−1
L−1
k=0
2L − 1
k
η
k
,l=0
1
(1 + η)
2L−1
L−1−l
k=0
2L − 1
k
b
a
l
η
k
−
a
b
l
η
2L−1−k
,l=0
PROBA B I L I T Y EXPRESSION
Pr{D<0}
Q
1
(a, b)+
L−1
l=0
C
l
(a, b, η)I
l
(ab)exp
−
1
2
(a
2
+ b
2
)
matrix of the vector x
k
as
m
k
=
ˆ
h
k
s
ˆ
h
k
, R =
χ |s|
2
+ N
0
χs
χs
∗
χ
. (10)
Using Table I expressions, equations (8) and (10), and
taking into account
N
R
k=1
|
ˆ
h
k
|
2
=
ˆ
λ, the constants a, b and η
are given by
a = a
n
ˆ
λ, with
a
n
=
1+2κ
n
¯γχ − 2
κ
n
¯γχ(κ
n
¯γχ +1)
2(κ
n
¯γχ +1)χ
,
b = b
n
ˆ
λ, with
b
n
=
1+2κ
n
¯γχ +2
κ
n
¯γχ(κ
n
¯γχ +1)
2(κ
n
¯γχ +1)χ
,
η = η
n
=1+2κ
n
¯γχ +2
κ
n
¯γχ(κ
n
¯γχ +1),
(11)
where the average SNR is defined as
γ
E
S
N
0
, and the constant
κ
n
3(2n−1)
2
2(M−1)
for M-QAM and κ
n
3(2n−1)
2
(L
2
−1)
for L-PAM.
Note that, after some algebra manipulations, it can be shown
that η
n
= b
n
/a
n
.
Substituting the parameters a, b,andC
l
in the expression
of P{D<0} that appears in Table I, the CCEP is expressed
as
I(n;
ˆ
h)=I(n;
ˆ
λ)=Q
1
(a
n
ˆ
λ, b
n
ˆ
λ)+
N
R
−1
l=0
C
l
(a
n
,b
n
)I
l
(a
n
b
n
ˆ
λ)exp
−
ˆ
λ
2
(a
2
n
+ b
2
n
)
,
(12)
where Q
1
is the Marcum-Q function, and I
l
is the l-th order
modified Bessel function of the first kind.
In the absence of channel prediction error, i.e. Ψ
k
=0,the
resultant noise after MRC n
, whose expression is given in (3),
is a Gaussian variable conditioned on
ˆ
h whose variance is
N
0
/
ˆ
λ. Thus, the calculation of CCEP is equivalent to the BER
calculation for a standard Gaussian noise channel, resulting in
I(n;
ˆ
λ)=Q
2
ˆ
λκ
n
¯γ
, (13)
where Q is the Gaussian Q-function.
B. Conditional and average BER expressions
The CCEP calculated in the previous section allows us to
obtain the conditional BER (CBER). This probability repre-
sents the BER conditioned on the predicted effective channel
gain
ˆ
λ, i.e, the BER under imperfect channel state information
(CSI). Introducing (12) in (4) yields
CBER(
ˆ
λ)=
L−1
n=1
ω(n)I(n;
ˆ
λ). (14)
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BER =
L−1
n=1
N
1
k=1
(N
2
+N
1
−2k)k
r=N
2
−N
1
ω(n)B
k,r
k
r+1
×
r!
k
r+1
1+
b
2
n
s
n,k
r
m=0
(m +1)
2k
s
n,k
m
a
2
n
s
n,k
2
F
1
m +2
2
,
m +2
2
+
1
2
;2;
4a
2
n
b
2
n
s
2
n,k
−
1
1+m
2
F
1
m +1
2
,
m +1
2
+
1
2
;1;
4a
2
n
b
2
n
s
2
n,k
!
+
N
R
−1
l=0
(r + l)!
(a
n
b
n
)
r+1
1
l!
w
n,k
− 1
w
n,k
+1
l/2
w
n,k
+1
2
r
×
2
F
1
−r, −r + l; l +1;
w
n,k
− 1
w
n,k
+1
"
w
2
n,k
− 1
r+1
!
,
(17)
The BER is obtained by averaging the CBER over the pre-
dicted eigenvalue
ˆ
λ as
BER =
L−1
n=1
ω(n)
#
∞
0
I(n;
ˆ
λ)p(
ˆ
λ)d
ˆ
λ. (15)
Using the fact that the pdf of the largest eigenvalue of
complex Wishart matrices can be expressed as a weighted sum
of elementary Gamma pdfs [17], the pdf of
ˆ
λ is given by
p(
ˆ
λ)=
N
1
k=1
(N
2
+N
1
−2k)k
r=N
2
−N
1
B
k,r
ˆ
λ
r
(1 − χ)
r+1
exp
−k
ˆ
λ
1 − χ
,
(16)
where N
1
Δ
=min{N
T
,N
R
}, N
2
Δ
=max{N
T
,N
R
} and
the coefficients B
k,r
are easily deducted from the expression
[17, eq.10].
Substituting equations (12) and (16) in (15), using
the expression [18, eq.8.772-3] and the integrals given in
[18, eq.8.914-1] and [19], we can obtain, after some algebraic
manipulation, the closed-form BER expression (17) at the top
of the p age, where
2
F
1
is the Gauss hypergeometric function,
s
n,k
= a
2
n
+ b
2
n
+2k and w
n,k
=
1 −
4a
2
n
b
2
n
s
2
n,k
−1/2
.Note
that function
2
F
1
in (17) can be expressed as a finite sum of
elementary functions for the values of its arguments, although
it is not shown here for compactness reasons.
For th e perfect CSI case, substituting expression (13) and
(16) in (15) the BER under perfect CSI is obtained as
BER =
L−1
n=1
N
1
k=1
(N
2
+N
1
−2k)k
r=N
2
−N
1
ω(n)B
k,r
r!
2k
r+1
×
1 −
"
κ
n
¯γ
k + κ
n
¯γ
r
l=0
2l
l
k
4(k + κ
n
¯γ)
l
,
(18)
whereweusethefactthattherequiredintegralover
ˆ
λ is
formally identical to [9, Eq. 5.18].
IV. N
UMERICAL RESULTS
The numerical results are obtained from analytical expres-
sions and simulations assuming a prediction subsystem similar
to that described in [4]. We consider Jakes’ model for the
channel time-correlatio n and Wienner filtering to predict the
CSI. The numerical values for the system parameters are:
carrier frequency f
c
=3 GHz, mobile speed v=36 km/h, feed-
back delay τ=1.28 ms and frame interval T =0.64 ms, which
0 5 10 15 20 25
Average SNR
BER
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
L = 2
M =4
M =16
MI MO 2x2
PCSI
F=16
Simulation
γ
Fig. 1. BER as a function of average SNR γ for different constellations
under perfect CSI (PCSI) and imperfect prediction (F =16 taps).
e.g. could correspond to a system with a symbol frequency
f
S
=100 kHz and 64 symbols per frame.
Figure 1 shows the BER for a 2x2 MIMO system as a
function of the average SNR ¯γ for some L-PAM and M-QAM
constellations in two cases: perfect CSI using equation (18)
and imperfect CSI using (17) with a 16-tap FIR Wiener
prediction filter. In the last case, we assume channel estimation
based on pilot symbol assisted modulation (PSAM) and the
mean square error χ is computed as described in [4, section
II-B]. As a double check, simulation results for the imperfect
CSI case are also superimposed in the figure. There is an
unavoidable channel prediction error floor independent of the
number of filter taps and, thus, the BER for perfect CSI
cannot be achieved. Note that at around BER ≈ 10
−3
the
relative SNR losses due to the channel prediction error for
the considered system parameters are about 2.5 dB for BPSK,
1.5 dB for 4-QAM and less than 1 dB for 16-QAM.
InFigure2theinfluence of antenna configuration on
the BER is depicted f or 16-QAM. Obviously in the SISO
case the channel prediction error does not affect the BER
performance. Moreover, the 2x4 and 4x2 MIMO systems show
the same BER for perfect CSI. This is because, if perfect
CSI is assumed, the BER performance for different antenna
configurations only depends on the pdf of
ˆ
λ, which is the
same for the 2x4 and 4x2 MIMO systems. However, for the
imperfect CSI case the BER performance is better for 2x4
MIMO systems.
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0 5 10 15 20 25
Average SNR
BER
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
N
T
=1, N
R
=1
N
T
=2, N
R
=2
T
N
T
=4, N
R
=4
PCSI
F=16
N
T
=2, N
R
=4
N=4,N
R
=2
γ
4-QAM
Fig. 2. BER as a function of average SNR γ for different antenna
configurations under perfect CSI (PCSI) and imperfect prediction (F =16 taps).
V. C ONCLUSIONS
Exact closed-form BER expressions for MIMO beam-
forming with MRC systems under channel prediction errors
have been derived. These results allow us to analyze the
performance of L-PAM and square M-QAM using p ractical
estimation methods under Rayleigh fading. The system perfor-
mance has been analyzed for different scenarios with different
values of the mobile speed, number of p rediction filter taps,
adaptation d elay, number of transmitter and receiver antennas,
constellation size and average SNR. Our results indicate that
practical constraints imposed by channel prediction d o not
significantly degrade BER performance.
A
CKNOWLEDGMENT
This work is partially supported by the Spanish Government
under project TEC2007-67289/TCM and by AT4 wireless.
The work of A. Goldsmith was supported in part by LG
Electronics.
R
EFERENCES
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