Equal Gain MIMO Beamforming in the RF
Domain for OFDM-WLAN Systems
´Alvaro Gonzalo1, Ignacio Santamar´ ıa1, Javier V´ ıa1, Fouad Gholam1, and Ralf
1University of Cantabria, 39005 Santander, Spain
2Dresden University of Technology, 01062 Dresden, Germany
Abstract. Equal gain beamforming (EGB) schemes are typically ap-
plied in the baseband domain and hence require complex RF transceivers.
In order to simplify the circuitry and energy consumption of the MIMO
transceiver, in this paper we consider an EGB scheme that operates in
the RF domain by means of analog phase shifters. Under OFDM trans-
missions, the design of the optimal phases is a complicated nonconvex
problem with no closed-form solution. Building upon a previously pro-
posed solution for flat-fading MIMO channels, this paper describes an
alternating minimization algorithm to find an approximate (suboptimal)
solution for the OFDM case. Monte-Carlo simulations are performed in
order to demonstrate the effectiveness of this new analog beamforming
scheme under coded and uncoded WLAN 802.11a transmissions.
Key words: Analog Combining, Multiple-Input Multiple-Output (MIMO),
Equal-Gain MIMO Beamforming, Orthogonal Frequency Division Mul-
tiplexing (OFDM), Wireless Local Area Networks (WLAN).
Conventional multiple-input multiple-output (MIMO) systems require all an-
tenna paths to be independently acquired and jointly processed at baseband.
The hardware cost, complexity and power consumption are therefore increased
accordingly. These drawbacks might explain, at least partially, why MIMO tech-
nologies have not found yet widespread use in low-cost wireless terminals. One
way to increase the energy-efficiency of MIMO terminals and reduce their costs is
to simplify the associated hardware and radio-frequency (RF) circuitry as much
as possible, while still retaining some of the benefits provided by the MIMO
channel (e.g., spatial diversity) by means of specifically designed signal process-
ing algorithms. With this goal in mind, a RF-MIMO architecture that performs
spatial processing directly in the analog domain is currently being developed
within the EU-funded project MIMAX [1, 2].
The combining scheme considered in [1, 2], which is depicted for conve-
nience in Fig. 1, permits to change the amplitudes and phases of the trans-
mitted/received RF signals by means of vector modulators (VM). Therefore, for
flat-fading MIMO channels and assuming perfect channel state information at
2A. Gonzalo et. al.
both sides of the link, it can implement the optimal maximum ratio beamforming
(MRB) solution. For this reason, in this paper we will refer to this architecture
as RF-MRB (i.e., radio-frequency maximum ratio beamforming). A drawback of
Antenna 1 Antenna 1
Fig. 1. Maximum ratio beamforming in the radio-frequency domain (RF-MRB).
the RF-MRB topology is that the average power can vary widely across anten-
nas, which is undesirable for the amplifiers since it can decrease their efficiency
. In order to mitigate this problem, in this paper we investigate an alternative
radio-frequency equal gain beamforming (RF-EGB) scheme, which substitutes
the vector modulators along each branch by analog phase shifters. Specifically,
we focus on the optimization problem that results from this beamforming archi-
For flat-fading single-input multiple-output (SIMO) or multiple-input single-
output (MISO) channels, the equal gain beamformers that maximize the signal-
to-noise (SNR) ratio are given by the phases of the SIMO or MISO channel,
respectively . For flat-fading MIMO channels, however, the optimization prob-
lem is nonconvex and no closed-form solution is known. Recently, Zheng et. al.
have proposed in  an alternating minimization algorithm for the flat-fading
MIMO case that uses the SIMO and MISO closed-form solutions iteratively by
fixing one side of the link and solving for the other. Under OFDM transmis-
sions the optimization problem becomes more challenging, since now we have to
optimize a global measure of performance (typically the SNR) using a common
set of Tx-Rx phases for all subcarriers. Building upon  and our own previous
work in [6, 7, 8], the main contribution of this paper is to provide a suboptimal
solution for this optimization problem and study its performance by means of
This paper is organized as follows. In Section 2 we present the analog MIMO
beamforming architecture based on phase shifters. In Section 3 we summarize the
EGB algorithm for flat-fading MIMO channels proposed in . Section 4 contains
the main contribution of this paper, which is the approximate maximum SNR
solution for the RF-EGB architecture under OFDM transmissions. In Section
5 we compare the performance in 802.11a WLAN systems of the proposed RF-
EGB beamforming architecture with the RF-MRB, the full-baseband MIMO
Equal Gain RF-MIMO Beamforming3
and the SISO schemes. Finally, the main conclusions are summarized in Section
Bold upper and lower case letters denote matrices and vectors respectively; light-
faced lower case letters denote scalar quantities. We use (·)H, (·)Tand ?·? to
denote Hermitian, transpose and the Frobenius norm, respectively. vmax(A) is
the principal eigenvector of the Hermitian semidefinite positive matrix A. We
use dist(x,y) to denote the Euclidean distance between vectors x and y. The
vector formed by the phase angles of x is denoted as ∠x. Finally, the expectation
operator is denoted as E [·].
2 Proposed RF-EGB architecture
The RF-EGB MIMO architecture studied in this paper is schematically shown in
Fig. 2. Essentially, the vector modulators in Fig. 1 are now substituted by wide-
band analog phase shifters. In this way, we avoid the power imbalance among
the various antenna branches and the wide power variations that can happen
in maximum ratio beamforming schemes. As long as the gains of the amplifiers
of the various branches match, the rest of the parameters can be relaxed and
inexpensive amplifiers can then be utilized. We consider WLAN 802.11a trans-
Antenna 1Antenna 1
Fig. 2. Equal gain beamforming in the radio-frequency domain (RF-EGB).
missions  that use orthogonal frequency division multiplexing (OFDM) and
achieve a data transmission of up to 54 Mbps. It is important to mention here
that the RF-EGB architecture does not try to solve the PAPR (peak-to-average
power ratio) problem of OFDM modulations, it just avoids power variations
among the analog signal paths. To mitigate this important problem of OFDM
systems, we could apply one of the many proposed PAPR reduction techniques
 or operate the amplifiers with some back-off.
4 A. Gonzalo et. al.
It is also assumed that both the transmitter and the receiver have perfect
channel state information, which has been obtained using specifically designed
training sequences. The system is intended for low-mobility indoor scenarios
as those encountered in WLAN transmissions, therefore we consider that the
channel remains static during the transmission of several frames. More details
about the training procedure and other implementation aspects of RF-MIMO
transceivers can be found in [1, 2] and the references therein.
The RF-EGB architecture poses several implementation challenges. For in-
stance, it can be difficult to design wideband phase shifters achieving a constant
phase change without any amplitude variation over the 20 MHz bandwidth re-
quired in WLAN 802.11a transmissions. Also, the automatic gain control system
at the receiver side can affect the equal gain beamforming. To simplify the analy-
sis, however, in this paper we will consider an idealized system in which all these
circuitry impairments are neglected, and we will focus just on the optimization
3 RF-EGB solution for flat-fading MIMO channels
In this section we describe the baseband model for MIMO beamforming schemes,
and summarize the iterative EGB solution proposed in  for flat fading MIMO
channels. Let us consider an Nr× NtMIMO system with Nttransmit and Nr
receive antennas. The unit-norm transmit and receive beamformers are wt =
(wt,1wt,2...wt,Nt)Tand wr = (wr,1wr,2...wr,Nr)T, respectively. Although these
beamformers are implemented in the RF domain, we can use the conventional
baseband model for the received signal
y = wH
rHwts + wH
where s ∈ C is the transmitted symbol, H ∈ CNr×Ntis the flat-fading MIMO
channel matrix, and n ∈ CNr×1is the noise vector, whose entries are independent
identical distributed (i.i.d.) complex Gaussian random variables with zero-mean
and variance σ2
Notice that using MIMO beamforming the symbols are transmitted through
an equivalent SISO channel: h = wH
rHwt. Assuming now that the transmitted
sequence has unit power, the receive SNR is given by
With maximum ratio beamforming we obtain the beamformer weights (am-
plitudes and phases) that maximize the receive SNR given by (1). As it is well-
known, this maximization problem has a closed-form solution which is given by
the left and right singular vectors corresponding to the largest singular value
of H . However, for equal gain beamforming an additional constraint must
Equal Gain RF-MIMO Beamforming5
be added to the problem: the elements of the transmit and receive beamformers
have a constant modulus, 1/√Ntfor wt, and 1/√Nrfor wr. Therefore, the SNR
maximization problem with EGB can be formulated as
subject to : |wr,i|2=
Nr,i = 1,2,...,Nr,
j = 1,2,...,Nt.
This is a nonconvex optimization problem which has no known closed-form
solution. In  this problem is solved by means of a cyclic algorithm. First, it is
shown that, unlike the MIMO case, for MISO and SIMO channels the equal gain
beamforming problems have the following well-known closed-form solutions 
where hMISO∈ C1×Ntand hSIMO∈ CNr×1are the MISO and SIMO channel vec-
tors, respectively. By exploiting this result, the equal gain beamformers for the
MIMO case are obtained in  applying an alternating minimization approach
1. Step 0: Initialize wr as the left singular vector of H corresponding to its
largest singular value.
2. Step 1: Fix wrand obtain wtas the solution of the MISO case by taking
hMISO= wrH as the effective (equivalent) MISO channel
3. Step 2: Fix wtand obtain wras the solution of the SIMO case by taking
hSIMO= Hwtas the effective SIMO channel
Iterate steps 1 and 2 until a given stop criterion is satisfied, in our case we
proposed to use the Euclidean distance between two consecutive beamformers
as the stop criterion. Specifically, the algorithm is stopped when the following
two conditions are simultaneously satisfied:
|dist(wt,k,wt,k−1)| < distmax
where k denotes iteration and distmax is the maximum error allowed. As an
example of its performance, we show in Fig. 3 a comparison between MRB and
EGB for a 4x4 Rayleigh flat-fading MIMO channel. It is clear that EGB attains
the same spatial diversity than MRB, although we are losing part of the array
gain, about 1.2 dB in this particular example.
|dist(wr,k,wr,k−1)| < distmax, (6)
6 A. Gonzalo et. al.
Fig. 3. Bit error rate comparison for EGB and MRB. QPSK symbols.
4 RF-EGB solution for OFDM-MIMO channels
Under OFDM transmissions, the RF-EGB optimization problem becomes even
harder due to the coupling among subcarriers. In fact, this problem is closely re-
lated to the design of pre-FFT schemes, which have been proposed to reduce the
computational cost of conventional OFDM-MIMO transceivers [12, 13]. However,
these pre-FFT techniques are applied in the baseband and typically optimize the
amplitudes and phases, therefore they are not directly applicable to our system.
Assume an OFDM transmission scheme with Nc data carriers and with a
cyclic prefix longer than the channel impulse response and let Hk∈ CNr×Ntbe
the MIMO channel for the k-th data-carrier. After analog Tx-Rx beamforming,
at each carrier we have an equivalent SISO channel given by
rHkwt,k = 1,...,Nc.
Our goal is to find the equal gain Tx-Rx beamformers maximizing the overall
receive SNR, i.e.,
Therefore we can pose the RF-EGB SNR maximization problem as follows
subject to : |wr,i|2=
i = 1,2,...,Nr
|wt,j|2=j = 1,2,...,Nt.
Equal Gain RF-MIMO Beamforming7
To obtain a suitable solution to this problem, we suggest a cyclic algorithm
inspired by the one previously described for the flat-fading case in Section 3. In
the next subsection we first propose a closed-form (but suboptimal) solution for
the frequency-selective MISO/SIMO cases.
4.1 EGB for frequency-selective MISO/SIMO channels
For a MISO channel, the problem (8) is reduced to
subject to : |wt,i|2=
Nt,i = 1,2,...,Nt
where hMISOk∈ C1×Ntis the MISO channel vector for the k-th carrier. In order
to derive a suboptimal solution for this problem let us rewrite (9) as
subject to : |wt,i|2=
Nt,i = 1,2,...,Nt
Again, the max-SNR problem for the MISO case in Eq. (10) is a complicated
nonconvex problem with no closed-form solution. In this paper we propose to
use the following simple, yet accurate, approximate solution given by the phases
of the principal eigenvector of RMISO
This solution is motivated by the fact that the main eigenvector of RMISOcon-
tains most of the channel energy averaged across carriers and, in consequence, its
phases should be close to the optimal solution of (10). In the simulation section
we will show some results supporting this claim.
Analogously, the solution for the SIMO case is given by
8A. Gonzalo et. al.
4.2 Alternating minimization algorithm
Inspired by the cyclic algorithm in , which was summarized in Section 3, we
solve the SNR maximization problem in (8) as follows
1. Step 0: Initialize wr(e.g., to a random value).
2. Step 1: Consider wr fixed and take hMISOk= wH
MISO channel for each subcarrier. The solution for wtis then given by (11).
3. Step 2: With wtfixed to the value obtained in the previous step, construct
the equivalent SIMO channels as hSIMOk= Hkwtand obtain the solution
for the receive equal gain beamformer as (12).
rHk as the equivalent
Steps 1 and 2 are iterated until the criterion defined in (6) is satisfied.
5 Simulation Results
In this section, we evaluate the performance of the proposed algorithm by means
of numerical simulations in MATLAB. For all simulations we consider a Rayleigh
4x4 MIMO channel model with an exponential power delay profile parameterized
by ρ, i.e.,
where L = 16 is the length of the channel impulse response which is assumed to
be equal to the cyclic prefix length. To check the convergence of the algorithm
we use a maximum error of distmax= 10−3(see Eq. (6)).
To verify the goodness of our method we first study how far is the proposed
suboptimal solution from the optimal one obtained by means of a brute-force
search. To this end we consider simple 1 × 2 and 1 × 3 MISO systems and
evaluate the energy of the equivalent SISO channel,?Nc
(i.e., ρ = 1), we found that the optimal equal-gain beamformer outperforms
our suboptimal solution by less than 10−3dB and 10−2dB for the 1 × 2 and
1 × 3 MISO channels, respectively. In addition, we found that as the channel
becomes less frequency selective our suboptimal solution gets even closer to the
optimal one. This support our claim that the proposed method provides a good
approximation of the optimal equal-gain beamforming phases.
Now, we compare the performance of the following methods:
= ρl1 − ρ
1 − ρLNrNt
l = 0,...,L − 1,
k=1| hMISOkwt|2, for
both the suboptimal and optimal solutions. For high frequency-selective channels
1. RF-MIMO architecture with the proposed equal gain beamforming algo-
2. RF-MIMO architecture with maximum ratio beamforming (RF-MRB)
3. Conventional baseband MIMO-OFDM system with optimal (per-carrier)
maximum ratio beamforming (Full-MIMO)
4. SISO system.
Equal Gain RF-MIMO Beamforming9
For the RF-MRB architecture we apply the maximum SNR solution obtained
with the algorithm described in . The Full-MIMO and the SISO schemes can
be seen as the upper and lower bounds, respectively, for the performance of any
system. For these simulations ρ has been fixed as ρ = 0.7, which represents a
high frequency-selective MIMO channel.
We consider coded and uncoded transmissions with frames generated accord-
ing to the 802.11a WLAN standard  (OFDM symbols with 64 carriers, out
of them 48 are data carriers, 4 are pilots and the rest are unused). Neverthe-
less, let us remark that throughout this paper we have assumed perfect channel
knowledge and therefore the pilots were not used for channel estimation. The
impact of the channel estimation errors and other RF impairments is left for
In the first simulation example we consider uncoded QPSK modulated data
to be transmitted over the 48 data carriers. The bit error rate (BER) curves
for the different methods are shown in Fig. 4. As we can observe, for uncoded
Fig. 4. Bit error rate vs. SNR for the compared algorithms. Uncoded QPSK symbols.
transmissions and high frequency-selective MIMO channels, both RF-MIMO ar-
chitectures fail to extract any or almost no frequency/spatial diversity, since the
performance is limited by the worst subcarriers. Both schemes, however, achieve
an important array gain in comparison to a SISO system. The RF-EGB perfor-
mance is always inferior to the RF-MRB performance and the gap depends on
the number of antennas and the frequency selectivity of the channel (i.e., ρ). For
a BER=10−3this loss is approximately 1.8 dB.
For the second example we have chosen a more realistic scenario using coded
transmissions under the 802.11a standard for a 12 Mbps rate (QPSK modulation
and a code rate of 1/2). The data bits are encoded with a convolutional code
and block interleaved as specified in the 802.11a standard. The receiver is based
on a hard decision Viterbi decoder. The results are shown in Fig. 5: with coded
10 A. Gonzalo et. al.
transmissions, both analog combining schemes are able to extract, at least partly,
the frequency and spatial diversity of the channel, although there is still an
important gap with respect to the Full-MIMO architecture. Nevertheless, as it
has been shown in [6, 7, 8] this gap can be diminished by optimizing other cost
functions (the mean square error, for instance) instead of the SNR. Obviously,
MRB achieves better results than EGB; however, the difference for a BER=10−3
is about 1.4 dB. Again, this supports our claim that the approximate EGB
solution proposed in this paper is close to the optimal one.
Fig. 5. Bit error rate for the compared algorithms. Coded QPSK symbols, R=1/2, 12
In Fig. 6 we illustrate the convergence of the proposed alternating mini-
mization algorithm for different values of ρ. These plots represent the Euclidean
distance between beamformers obtained in two consecutive iterations. The con-
vergence curves have been obtained by averaging 500 independent trials, and the
vertical bars indicate the variance. Finally, Fig. 7 presents the evolution of the
BER curves with the number of iterations in the SNR range of 10-13 dB. We
can conclude that the algorithm converges very fast within the first 10 iterations,
although the convergence speed decreases and the variance increases for larger
values of ρ (i.e., for high frequency-selective channels).
MIMO transceivers performing spatial processing in the RF domain (RF-MIMO)
are a good alternative in order to reduce the hardware cost and system size, as
well as to increase the energy-efficiency of the system. In this paper we have
studied a particular scheme (RF-EGB), which applies the equal gain combining
concept and only uses phase shifters before the analog combiner, instead of full
Equal Gain RF-MIMO Beamforming 11
Fig. 6. Convergence of the algorithm for different values of ρ. 4x4 antenna configura-
Fig. 7. BER curves for different number of iterations. 4x4 RF-EGB system, QPSK
uncoded symbols and channel with ρ = 0.7.
vector modulators as previously proposed (RF-MRB scheme). Under OFDM-
WLAN transmissions, the proposed scheme results in a complicated optimization
problem since the Tx-Rx analog equal gain beamformers simultaneously affects
all subcarriers. We have proposed simple (but suboptimal) solutions for the
MISO and SIMO cases, and based on these, a cyclic minimization algorithm to
get the maximum SNR solution for the MIMO case. The proposed algorithm has
been shown to provide good results with a low computational complexity, since it
converges in very few iterations. Coded and uncoded data 802.11a transmissions
have been simulated, and in both cases, RF-EGB has shown to behave only
slightly inferior to the RF-MRB scheme.
12 A. Gonzalo et. al.
The research leading to these results has received funding from the European
Community’s Seventh Framework Programme (FP7/2007-2013) under grant
agreement n 213952, MIMAX. The work of the fourth author has been sup-
ported by MAEC-AECID (Agencia Espa˜ nola de Cooperaci´ on Internacional para
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