A General Pre-FFT Criterion for MIMO-OFDM Beamforming.

Abstract

In this paper, we propose a general beamforming criterion for pre-FFT processing in orthogonal frequency division multiplexing (OFDM) systems with multiple transmit and receive antennas. The proposed criterion depends on a single parameter α, which establishes a tradeoff between the energy of the equivalent SISO channel (after Tx-Rx beamforming) and its spectral flatness. The proposed cost function embraces most reasonable criteria for designing Tx-Rx pre-FFT beamformers. Hence, for particular values of α the proposed criterion reduces to the minimization of the mean square error (MSE), the maximization of the system capacity, or the maximization of the received signal-to-noise ratio (SNR). In general, the proposed criterion results in a non convex optimization problem. However, we show that the problem can be approximately solved by semidefinite relaxation (SDR) techniques. Additionally, since the computational cost of SDR for this problem is rather high, we propose a simple yet efficient gradient search algorithm which provides satisfactory solutions with a moderate computational cost for OFDM-based WLAN standards such as 802.11a. Finally, the good performance of the proposed technique is illustrated by means of some numerical results.

Full text

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A General Pre-FFT Criterion for
MIMO-OFDM Beamforming
Javier V´ ıa, Ignacio Santamar´ ıa and Victor Elvira
Dept. of Communications Engineering
University of Cantabria
39005 Santander, Cantabria, Spain
Email: {jvia,nacho,victorea}@gtas.dicom.unican.es
Ralf Eickhoff
Chair for Circuit Design and Network Theory
Dresden University of Technology
01062 Dresden, Germany
Email: Ralf.Eickhoff@tu-dresden.de
Abstract—In this paper, we propose a general beamforming
criterion for pre-FFT processing in orthogonal frequency division
multiplexing (OFDM) systems with multiple transmit and receive
antennas. The proposed criterion depends on a single parameter
α, which establishes a tradeoff between the energy of the equiv-
alent SISO channel (after Tx-Rx beamforming) and its spectral
flatness. The proposed cost function embraces most reasonable
criteria for designing Tx-Rx pre-FFT beamformers. Hence, for
particular values of α the proposed criterion reduces to the
minimization of the mean square error (MSE), the maximization
of the system capacity, or the maximization of the received signal-
to-noise ratio (SNR). In general, the proposed criterion results in
a non-convex optimization problem. However, we show that the
problem can be approximately solved by semidefinite relaxation
(SDR) techniques. Additionally, since the computational cost of
SDR for this problem is rather high, we propose a simple yet
efficient gradient search algorithm which provides satisfactory
solutions with a moderate computational cost for OFDM-based
WLAN standards such as 802.11a. Finally, the good performance
of the proposed technique is illustrated by means of some
numerical results.
I. INTRODUCTION
In the last years, pre-FFT beamforming schemes have
emerged as an interesting alternative for exploiting spatial
diversity in multiple-input multiple-output (MIMO) systems
under orthogonal frequency division multiplexing (OFDM)
transmissions [1]–[3]. The key idea behind pre-FFT processing
consists in combining the received (respectively transmitted)
signals before (resp. after) FFT processing, thus avoiding the
necessity of one FFT block for each antenna (see Fig. 1). An
equivalent scheme appears when beamforming is performed
in the analog domain [4], [5].
The price to be paid by this reduction in the system com-
plexity is a slight performance degradation in comparison to
a full MIMO system that uses optimal per-carrier processing.
Obviously, this limitation comes from the fact that the same
pair of transmit/receive beamformers is applied to all the
subcarriers.
The research leading to these results has received funding from the Euro-
pean Community’s Seventh Framework Programme (FP7/2007-2013) under
grant agreement n 213952, MIMAX. Additionally, the work of the first three
authors has also been supported by the Spanish Government (MICINN) under
projects TEC2007-68020-C04-02/TCM (MULTIMIMO) and CONSOLIDER-
INGENIO 2010 CSD2008-00010 (COMONSENS).
?
MIMO?
beamforming
weights
?
?
T
w
R
w
Insert/Remove
cyclic prefix
IFFT/FFT
Tx?
Rx
Encoded
symbols
Fig. 1.Pre-FFT MIMO beamforming scheme.
Previous works have considered the problem of designing
the pre-FFT beamformers in order to maximize the received
SNR [1]–[4], and more recently, we have proposed two new
beamforming criteria based on the maximization of the system
capacity [6], and the minimization of the bit error rate (BER)
of the optimal linear receiver [7]. In this work, we propose
a general pre-FFT beamforming criterion which includes, as
particular cases, all the previous approaches. The proposed
cost function is inspired by the Renyi entropy definition,
and it depends on a single parameter α, which establishes
a tradeoff between the energy of the equivalent channel
(after Tx-Rx beamforming) and its spectral flatness. Thus,
for particular values of this parameter, the proposed criterion
reduces to the maximization of the SNR (MaxSNR, α = 0),
the maximization of the system capacity (MaxCAP, α = 1),
the minimization of the MSE of the optimal linear receiver
(MinMSE, α = 2), and the optimization of the worst subcarrier
response (MaxMin, α = ∞).
Unfortunately, the proposed criterion results in a non-convex
optimization problem. However, we show that approximated
solutions can be obtained by means of semidefinite relaxation
(SDR) techniques. Furthermore, in order to avoid the high
computational cost of SDR approaches for practical problems,
we propose a simple iterative algorithm which, equipped with
a good initialization method, provides satisfactory results in
most practical cases.
978-1-4244-6404-3/10/$26.00 ©2010 IEEE
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II. DATA MODEL
Let us consider a pre-FFT MIMO system with nT transmit
and nR receive antennas, and with unit-energy transmit and
receive beamformers. Assuming a cyclic prefix longer than the
channel impulse response, the communication system after Tx-
Rx beamforming may be decomposed into the following set of
parallel and non-interfering single-input single-output (SISO)
equivalent channels
yk= hksk+ nk,k = 1,...,Nc,
(1)
where Nc is the number of data carriers, yk ∈ C is the
observation associated to the k-th data carrier, nk represents
the complex circular i.i.d. Gaussian noise with zero mean
and variance σ2, sk is the transmitted signal, and hk is the
frequency response of the equivalent channel after Tx-Rx
beamforming, which is given by
hk= wH
RHkwT,k = 1,...,Nc,
(2)
where wT ∈ CnT×1and wR ∈ CnR×1are the transmit
and receive beamformers, and Hk∈ CnR×nTrepresents the
response of the MIMO channel for the k-th data-carrier.
Under perfect knowledge of the equivalent channel and
noise variance, and assuming unit transmit power per data
carrier (E[|sk|2] = 1), the linear minimum mean square error
(LMMSE) estimate of sk is ˆ sk =
per-carrier MSE
?
where γ = 1/σ2is defined as the (expected) signal to noise
ratio (SNR) at the transmitter side.
h∗
kyk
|hk|2+σ2, which yields a
MSEk= E
|ˆ sk− sk|2?
=
1
1 + γ|hk|2,k = 1,...,Nc,
(3)
III. GENERAL PRE-FFT MIMO BEAMFORMING
CRITERION
In this section we introduce a general criterion for the
design of the Tx-Rx beamformers under perfect knowledge
of the MIMO channel Hkand noise variance at the receiver
side.1Specifically, we propose to minimize the following cost
function
?
Nc
fα(wT,wR) =
1
α − 1log
1
Nc
?
k=1
MSEα−1
k
?
, 0 ≤ α < ∞
(4)
which is inspired by the definition of Renyi’s entropy of
order α for a discrete random variable [9], [10]. Thus, our
optimization problem can be written as
argmin
wT,wR
fα(wT,wR)
subject to
?wT? = ?wR? = 1,
(5)
which encompasses the following interesting beamformer de-
sign criteria:
• MaxSNR (α = 0): If the parameter α is set to zero, the
optimization problem in (5) reduces to the maximization
1Some numerical results including the estimation of the channel and noise
variance can be found in the journal version of this paper [8].
of the energy of the equivalent channel, or equivalently,
to the maximization of the received SNR. [1]–[4].
• MaxCAP (α = 1): When α approaches 1, it can be easily
shown by direct application of the L’Hopital’s rule, that
the proposed criterion reduces to the maximization of the
system capacity [6].
• MinMSE (α = 2): In this case, (5) is equivalent to the
minimization of the overall MSE of the optimal linear
receiver [7].
• MaxMin (α = ∞): In this case, the summation in (4)
is dominated by the worst data-carrier, i.e., by that with
the smallest |hk|2. Therefore, for α → ∞, the proposed
criterion reduces to the optimization of the worst data-
carrier.
A. Cost Function Properties
In this subsection, we briefly review the main properties
of the proposed beamforming criterion. For further details, as
well as a rigorous proof of the properties, we refer to [8].
Property 1: In the low SNR regime (γ → 0), the proposed
criterion reduces to the MaxSNR approach regardless of α.
Property 2: The solutions (˜ wT, ˜ wR) of the proposed beam-
forming criterion are Pareto optimal points of the multiob-
jective optimization problem based on the individual channel
energies (|hk|2) or MSEs (MSEk).
Property 1 ensures the equivalence of all the criteria in the
low SNR regime. Property 2 implies that, given the optimal
pair of beamformers (˜ wT, ˜ wR) for some value of α, then there
does not exist another feasible pair (wT, wR) satisfying
|wH
RHkwT|2≥ |˜ wH
with at least one strict inequality. However, we must note
that the converse of Property 2 is not true in general. This
fact is illustrated by means of a toy example consisting of
a SIMO system with Nc= 2 data carriers, nR= 2 receive
antennas, real beamformers and subcarrier channels given by
H1= [1,0]Tand H2= [0.35,−0.8]T.
Fig. 2 shows the set of achievable points (|h1|2,|h2|2) with
unit energy beamformers wR, where we can see that the
solutions of the proposed criterion for different values of α
are only a subset of the Pareto optimal solutions. However, it
should be pointed out that those points in the Pareto boundary
that are not achievable by our criterion might not be of
practical interest. As an example, consider the Pareto points
in Fig. 2 near |h1|2= 0.4, |h2|2= 0.7. Clearly, these points
are worse solutions than those near |h1|2= 0.8, |h2|2= 0.4,
which are solutions of the proposed criterion for some α ? 0.
The analogy of the proposed cost function with Renyi’s
entropy allows us to to shed some light into the effect of the
parameter α. Specifically, when α increases from 0 to ∞, the
proposed beamforming criterion tends to flatten the equivalent
channel, i.e., the critical data carriers (those with the smallest
|hk|) are improved at the expense of the rest. In other words,
we are sacrificing part of the channel energy in order to obtain
less frequency selective equivalent SISO channels.
RHk˜ wT|2,k = 1,...,Nc,
(6)
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0 0.2 0.40.6 0.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
|h1|2
|h2|2
Achievable Points
Pareto Optimal Points
Proposed Solutions
|h2|2=C?|h1|2
|h1|2=|h2|2
?=?
?=3
?=2
?=1
?=0
These points are worse
than those near the
MaxSNR (?=0) solution
Fig. 2. Toy example with a 1×2 SIMO system with Nc= 2 subcarriers. The
figure shows the set of achievable points with ?wR? = 1 (dotted line), the
Pareto optimal points (solid line), and the solutions associated to the proposed
criterion (the section of the curve marked with circles).
TABLE I
PARTICULAR CASES WITH CLOSED-FORM SOLUTIONS.
System
wT
1
MRT
MRT
wR
MRC
1
MRC
SIMO with α = 0 or γ ? 0
MISO with α = 0 or γ ? 0
MIMO flat fading
IV. OPTIMIZATION PROBLEM AND PROPOSED ALGORITHM
Although the optimization problem (5) is in general non-
convex, in this section we show that approximate solutions
can be obtained by means of semidefinite relaxation (SDR)
techniques. However, the computational cost associated to
SDR techniques can be very high even for a moderate number
of data subcarriers and antennas. For this reason, we also
propose a simple gradient search method which is initialized
using a closed-form approximation of the MaxSNR solution.
As it will be shown in Section V, this method provides very
accurate results.
A. Optimization Problem
Let us start by rewriting the optimization problem in (5) as
?
Nc
k=1
argmin
W
1
α − 1log
1
Nc
?
?
1 + γ |Tr(HkW)|2?1−α?
,
(7)
subject to
?W? = 1,
rank(W) = 1,
where W = wTwH
matrix. Although the solution of the above problem can be
obtained in closed-form in some particular cases (see Table I),
in general this is a very difficult problem due to the rank-one
constraint and the non-convexity of the cost function, which
precludes the application of standard convex optimization
techniques [11]. Nevertheless, we can gain some insight by
Ris the rank-one Tx-Rx beamforming
applying the Lagrange multipliers method. In particular, it is
easy to prove that the local minima of (7) are also solutions
of the following coupled eigenvalue (EV) problems
RMISOαwT= λwT,
where λ =?Nc
RMISOα=
RSIMOαwR= λwR,
(8)
k=1MSEα
k|hk|2,
Nc
?
Nc
?
k=1
MSEα
khMISOkhH
MISOk,
(9)
RSIMOα=
k=1
MSEα
khSIMOkhH
SIMOk,
(10)
can be seen as weighted covariance matrices, and
hMISOk= HH
kwR,
hSIMOk= HkwT,
(11)
are the MISO (SIMO) channels after fixing the receive (trans-
mit) beamformer. Thus, the proposed beamforming criterion
tries to maximize a weighted-sum of the energies of the
equivalent SISO channel, where the weights are precisely
given by MSEα
the worst subcarriers.
kand hence the higher weights are given to
B. Approximated Solution based on Semidefinite Relaxation
In order to obtain an approximate solution to our non-
convex optimization problem, it has to be reformulated in
a form suitable for the application of semidefinite relaxation
(SDR) techniques. Thus, it is not difficult to prove that (7) can
be rewritten as [8]
argmin
W,˜
W,γ1,...,γNc
subject to
1
α − 1
γTr(˜Hk˜ W) = γk,
Tr(˜ W) = 1,
vec(W) = vmax(˜ W),
rank(˜ W) = 1,
Nc
?
k=1
(1 + γk)1−α,
(12)
k = 1,...,Nc
˜ W ? 0,
rank(W) = 1,
where˜Hk= hkhH
SIMO or MISO channel with nTnRantennas, w = vec(W),
and ˜ W = wwHis the associated nTnR× nTnR rank-one
beamforming matrix. On the other hand, vmax(˜ W) denotes
the eigenvector corresponding to the maximum eigenvalue of
˜ W, and˜ W ? 0 means that˜ W is hermitian and semidefinite
positive.
Now, dropping the two rank-one constraints we obtain the
following convex optimization problem2
k, hk= vec(HH
k) can be seen as a virtual
argmin
˜
W,γ1,...,γNc
subject to
1
α − 1
γTr(˜Hk˜ W) = γk,
Tr(˜ W) = 1,
Nc
?
k=1
(1 + γk)1−α,
(13)
k = 1,...,Nc
˜ W ? 0,
2Note that the matrix W can be removed because it only appears in the
constraint vec(W) = vmax(˜ W).
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Page 4

which can be solved by means of standard convex optimization
techniques. Finally, an approximated rank-one solution to (12)
can be generated from the (in general full-rank) solution of
(13) by means of randomization [12] or a similar approach.
In general the computational complexity of the SDR tech-
nique can be prohibitive for practical applications. As an
example, let us consider a practical MIMO system such as
that used in the simulations (Nc = 64 data carriers and
nT = nR = 4 antennas). In this case, we have Nc = 64
slack variables γkand a 16 × 16 semidefinite positive matrix
˜ W. Thus, even assuming a linear cost function, the com-
putational cost of the problem in (13) can be as large as
O?(Nc+ n2
candidates (see [12] for typical numbers of randomizations
used in a moderate-sized problem) would also result in a
prohibitive computational burden.
Tn2
R)3.5?≈ 5 · 108[12]. Furthermore, the appli-
cation of a randomization method with a sufficient number of
C. Proposed Beamforming Algorithm
In order to avoid the high computational complexity asso-
ciated to the SDR approach, we propose a simple iterative al-
gorithm which, equipped with an adequate initialization point,
provides good results in most practical cases. The proposed
algorithm can be seen as a generalization of those proposed in
[6], [7] for the MaxCAP (α = 1) and MinMSE (α = 2) cases.
Let us start by briefly describing the initialization method,
which obtains an approximated MaxSNR solution in closed
form. In particular, for α = 0 (or γ ? 0) the optimization
problem in (7) can be rewritten as
argmax
W,w
Nc
?
?w? = 1,
vec(W) = w,
?Nc
k=1
wH˜Hkw,
(14)
subject to
rank(W) = 1.
Thus, defining R =
constraint, we can obtain an approximate solution of the above
problem as the principal eigenvector of R. As previously
pointed out, the matrix W obtained from the solution w =
vmax(R) will not be rank-one in general. Here, we propose to
obtain the best (in the squared-norm sense) rank-one approx-
imation of W, i.e., the transmit and receive beamformers are
given by the left and right singular vectors of W.
After obtaining the initialization point, the proposed gradi-
ent search algorithm is based on the following updating rules
k=1˜Hk and relaxing the rank-one
wT(t + 1) = wT(t) + μRMISOα(t)wT(t),
wR(t + 1) = wR(t) + μRSIMOα(t)wR(t),
(15)
(16)
where μ is a step-size (or regularization parameter) and t
denotes the iteration index.3The above expressions can be also
interpreted as iterations of a (regularized) power method for
3Analogously to conventional gradient-search methods, the convergence of
the proposed algorithm can be guaranteed by choosing a sufficiently small
value of µ. Obviously, this parameter also has a direct impact on the speed
of convergence, which increases with µ.
Select μ and α; initialize wT and wR (MaxSNR approximation).
repeat
Update of the transmit beamformer
Obtain the equivalent MISO channels hMISOkwith (11).
Update hk and MSEk for k = 1,...,Nc.
Obtain the matrix RMISOαwith (9).
Update the beamformer wT with (15).
Normalize the solution: wT = wT/?wT?.
Update of the receive beamformer
Obtain the equivalent SIMO channels hSIMOkwith (11).
Update hk and MSEk for k = 1,...,Nc.
Obtain the matrix RSIMOαwith (10).
Update the beamformer wR with (16).
Normalize the solution: wR = wR/?wR?.
until Convergence
Algorithm 1: Proposed beamforming algorithm.
obtaining the solution of the coupled EV problems in (8). The
overall technique, which includes a normalization step to force
the unit energy constraint on the beamformers, is summarized
in Algorithm 1.
Regarding the computational complexity, it is easy to ver-
ify that the initialization step has a complexity of order
O(n3
method comes at a cost of approximately O?Nc(nT+ nR)2?.
example (Nc = 64 and nT = nR = 4) would have a cost
three orders of magnitude lower than that of the SDR approach
previous to the randomization technique.
Tn3
R+ Ncn2
Tn2
R), whereas one iteration of the proposed
Thus, 50 iterations of the proposed algorithm in the previous
V. SIMULATION RESULTS
The performance of the proposed iterative technique is
illustrated in this section by means of some Monte Carlo
simulations.4In all the experiments, we consider a MIMO
system with 64 subcarriers and nT = nR = 4 transmit and
receive antennas. An i.i.d. Rayleigh MIMO channel model
with exponential power delay profile has been assumed. In
particular, the total power associated to the l-th tap is
?
where Lc is the length of the channel impulse response
(Lc= 16 in the simulations), and the exponential parameter
ρ has been selected as ρ = 0.7. We have focused on the
MaxSNR (α = 0) [1]–[4], MaxCAP (α = 1) and MinMSE
(α = 2) approaches, which have been compared with a SISO
system and with a full MIMO scheme applying maximum ratio
transmission (MRT) and maximum ratio combining (MRC)
per subcarrier (denoted in the figures as Full-MIMO), which
can be seen as an upper bound for the performance of any
pre-FFT system. Finally, in all the experiments, the step-size
has been fixed to μ = 0.1, and the convergence criterion is
based on the difference between the beamformers in two con-
secutive iterations. Specifically, the algorithm finishes when
the Euclidian distance is lower than 10−3. With these values,
the proposed algorithm has never exceeded 50 iterations.
E
?H[l]?2?
= (1−ρ)ρlnTnR,l = 0,...,Lc−1, (17)
4We have verified by means of simulations that the iterative algorithm
outperforms the SDR approach with a moderate number of randomizations.
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Page 5

05 10152025
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR (dB)
Poutage
SISO
Full?MIMO
MaxSNR (?=0)
MaxCAP (?=1)
MinMSE (?=2)
Fig. 3. Outage probability for a transmission rate of 5bps/Hz.
In the first experiment, we evaluate the outage probability
of the system for a transmission rate of 5bps/Hz, which is
shown in Fig. 3. As expected, the best results are provided by
the Full-MIMO system, followed by the pre-FFT MaxCAP
criterion (α = 1), which clearly outperforms the remaining
pre-FFT schemes and the SISO system.
The advantage of the MaxCAP and MinMSE criteria over
the MaxSNR approach becomes clearer when the system
performance is evaluated in terms of bit error rate (BER).
In particular, we have adopted the 802.11a standard, which
uses Nc = 48 out of 64 subcarriers for data transmission.
The information bits are encoded with a convolutional code
and block interleaved as specified in the standard. Finally,
the receiver is based on a soft Viterbi decoder, and we have
selected a transmission rate of 12Mbps, which implies QPSK
signaling and a convolutional code of rate 1/2. Here, the
introduction of a channel encoder could induce us to think
that the MaxCAP criterion will outperform the remaining
approaches. However, as can be seen in Fig. 4, the best results
are provided by the MinMSE beamformers. This is due to the
fact that we are not using an ideal channel encoder (note that
the channel encoder operates on an OFDM symbol basis),
which implies that a slight degradation in the capacity can
be acceptable in order to obtain a less frequency selective
equivalent channel.
VI. CONCLUSION
In this paper we have proposed a general MIMO beamform-
ing criterion for pre-FFT processing systems. It depends on a
single parameter, α, which establishes a tradeoff between the
energy and spectral flatness of the equivalent SISO channel.
Several interesting design criteria such as the MaxSNR, the
MaxCAP and the MinMSE are just particular cases of the
proposed criterion for different values of α. In general, the
proposed criterion results in a non-convex optimization prob-
lem, but approximated solutions can be obtained by means
of SDR techniques. Furthermore, in order to avoid the high
?10
?50510 15
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR (dB)
BER
SISO
Full?MIMO
MaxSNR (?=0)
MaxCAP (?=1)
MinMSE (?=2)
Fig. 4.
QPSK signaling and convolutional encoder of rate 1/2.
BER for a 802.11a based system with transmission rate of 12Mbps.
computational cost of SDR techniques, we have proposed a
simple and efficient algorithm which, with a proper initial-
ization, provides very good results in practical OFDM-based
WLAN standards such as 802.11a. Finally, we can conclude
that, in general, it is advisable to increase the spectral flatness
of the equivalent SISO channel, even at the expense of a slight
degradation of the overall SNR. Future research lines include
the extension of the presented criteria to multiuser scenarios.
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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 ICC 2010 proceedings