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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)