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Equal Gain MIMO Beamforming in the RF

Domain for OFDM-WLAN Systems

´Alvaro Gonzalo1, Ignacio Santamar´ ıa1, Javier V´ ıa1, Fouad Gholam1, and Ralf

Eickhoff2

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).

1 Introduction

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

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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

Base

band

Up

conversion

Weight

settling

Antenna Nt

Σ

Base

band

Down

conversion

Weight

settling

Antenna Nr

Antenna 1 Antenna 1

Transmitter Receiver

MIMO

CHANNEL

VM

(Amplitude

and phase)

VM

(Amplitude

and phase)

VM

(Amplitude

and phase)

VM

(Amplitude

and phase)

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

[3]. 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-

tecture.

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 [4]. 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 [5] 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 [5] 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

simulations.

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 [5]. 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

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Equal Gain RF-MIMO Beamforming3

and the SISO schemes. Finally, the main conclusions are summarized in Section

6.

1.1 Notation

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-

Base

band

Up

conversion

Weight

settling

Antenna Nt

Σ

Base

band

Down

conversion

Weight

settling

Antenna Nr

Antenna 1Antenna 1

TransmitterReceiver

MIMO

CHANNEL

Phase

shifter

Phase

shifter

Phase

shifter

Phase

shifter

Fig. 2. Equal gain beamforming in the radio-frequency domain (RF-EGB).

missions [9] 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

[10] or operate the amplifiers with some back-off.

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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

problem.

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 [5] 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

rn,

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

n.

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

???wH

E

|wH

SNR =

E

rHwts??2?

?

rn|2?

=

??wH

rHwt

σ2

n

??2

.(1)

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 [11]. However, for equal gain beamforming an additional constraint must

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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

??wH

subject to : |wr,i|2=

1

Nt,

max

{wr,wt}

rHwt

??2

(2)

1

Nr,i = 1,2,...,Nr,

|wt,j|2=

j = 1,2,...,Nt.

This is a nonconvex optimization problem which has no known closed-form

solution. In [5] 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 [4]

wt=ej∠hMISOH

√Nt

,wr=ej∠hSIMO

√Nr

. (3)

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 [5] applying an alternating minimization approach

as follows:

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

wt=ej∠HHwr

√Nt

. (4)

3. Step 2: Fix wtand obtain wras the solution of the SIMO case by taking

hSIMO= Hwtas the effective SIMO channel

wr=ej∠Hwt

√Nr

.(5)

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.

and

|dist(wr,k,wr,k−1)| < distmax, (6)

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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

hk= wH

rHkwt,k = 1,...,Nc.

Our goal is to find the equal gain Tx-Rx beamformers maximizing the overall

receive SNR, i.e.,

Nc

?

SNR =

k=1

??wH

rHkwt

??2

σ2

n

. (7)

Therefore we can pose the RF-EGB SNR maximization problem as follows

max

{wr,wt}

Nc

?

k=1

??wH

1

Nr,

1

Nt,

rHkwt

??2

(8)

subject to : |wr,i|2=

i = 1,2,...,Nr

|wt,j|2=j = 1,2,...,Nt.

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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

max

{wt}

Nc

?

k=1

| hMISOkwt|2

(9)

subject to : |wt,i|2=

1

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

max

{wt}wH

tRMISOwt

(10)

subject to : |wt,i|2=

1

Nt,i = 1,2,...,Nt

where

RMISO=

Nc

?

k=1

hMISOH

khMISOk.

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

wt=

1

√Nt

ej∠vmax(RMISO). (11)

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

wr=

1

√Nr

ej∠vmax(RSIMO),(12)

where RSIMO=?Nc

k=1hSIMOnhSIMOH

n.

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8A. Gonzalo et. al.

4.2 Alternating minimization algorithm

Inspired by the cyclic algorithm in [5], 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:

E

?H[l]?2?

= 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-

rithm (RF-EGB).

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.

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Equal Gain RF-MIMO Beamforming9

For the RF-MRB architecture we apply the maximum SNR solution obtained

with the algorithm described in [8]. 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 [9] (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

future work.

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

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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

Mbps.

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).

6 Conclusions

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

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Equal Gain RF-MIMO Beamforming 11

Fig. 6. Convergence of the algorithm for different values of ρ. 4x4 antenna configura-

tion.

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.

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12 A. Gonzalo et. al.

Acknowledgment

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

el Desarrollo).

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