1.1 Uplink and Downlink Bmf for Fading Channels

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    UPLINK AND DOWNLINK BEAMFORM ING

    FOR FADING CHANNELS

    Mats Bengtsson and Bj om O ttersten

    Signal Processing, S3, Royal Instituteof Technology

    100 44 Stockholm, SWEDEN

    Tel: 4 6 79084 63, fax: +46 8 790 72 60

    Email:

    matsb8s3.

    kth. e

    ABSTRACT

    We highlight some issues in the design of beamformers

    for transmission and reception in communications systems

    with antenna arrays at the base stations. We assume a ra-

    dio channel characterized by frequency flat Rayleigh fading

    which is correlated (but not coherent) from antenna element

    to antenna element.

    Different design criteria are considered and we show the

    relationship between downlink and uplink formulations and

    solutions. The resulting beamformers are typically given

    by quadratically constrained optimization problems which

    could be expected to give increased robustness to signal can-

    cellation. Unfortunately, signal cancellation is still present,

    but diagonal loading can be introduced to minimize the prob-

    lem.

    1.

    INTRODUCTION

    When antenna arrays are used at the base station of a cellu-

    lar system in environments with multipath propagation, the

    standard plane wave model used in much of the array pro-

    cessing literature is not applicable.

    As

    an alternative, some

    authors have used a fading channel where the fading pro-

    cess in independent between the different antenna elements

    [15,16] . Since the assumption of independently fading sub-

    channels only holds in environments with severe multipath

    or

    large separation between the antenna elements, we as-

    sume here a channel with correlated fading between the an-

    tenna elements. Such models can be used to model e.g. the

    diffuse scattering caused by reflections close to each mo-

    bile, see [ , 8,181. We assume that the channel is constant

    within the time frame allocated to one data burst but varies

    randomly from burst to burst.

    When the antenna array is used as a receiver, i.e. in

    uplink mode, the instantaneous channel can be estimated

    directly from the received data, whereas in the downlink,

    the transmitting beamformer must be based on information

    collected in the uplink. Several schemes have been pro-

    posed for the transformation from uplink to downlink. In a

    Time Division Duplex (TDD) system with sufficiently short

    time slots, the downlink channel

    is

    virtually identical to

    the uplink channel, whereas in a Frequency Division Du-

    plex (FDD) system, the channel fades independently at the

    two duplex frequencies. However, a statistical model of the

    downlink channel can be obtained from the collected up-

    link data using a physical model [5,18] or model-free tech-

    niques [2,9]. In this paper we assume that the statistics of

    the downlink channel is given, exactly or in the form of a

    noisy estimate.

    The maximum Signal to Interference plus Noise Ratio

    (SINR) solution has been used for different classes of prob-

    lems, see for example

    [171.

    For the model with local scat-

    tering, the uplink scenario is studied in

    [3]

    and the corre-

    sponding downlink formulation is given in [181. Here, we

    show that the maximum SINR solution can be interpreted as

    quadratically constrained minimum variance beamforming

    [7,12,13].

    Since the phase of the fading channel fluctuates ran-

    domly, a naive treatment of the Minimum Mean Square

    Er-

    ror (MMSE) problem gives the all-zero solution. Using the

    assumption of a coherent receiver, we give two different so-

    lutions to the problem. For the downlink, it is not obvious

    how to formulate an MMSE problem and interpret the re-

    sult. We propose to apply the uplink solution also in the

    downlink.

    Simulations have been performed to compare the dif-

    ferent beamformers and study the sensitivity to modeling

    errors. Diagonal loading

    is

    successfully used to avoid the

    problem of signal cancellation.

    2. DATAMODEL

    2.1.

    Uplink

    We assume a frequency flat Rayleigh fading channel, where

    the baseband data at the antenna array is collected in the

    complex valued

    x 1

    vector xu t )given by

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    k = l

    where Sk t) is the baseband signal transmitted at the kth

    mobile,

    n(t) s

    spatially and temporally white Gaussian noise

    with covariance matrix

    and the array response vector

    vk

    corresponding to mobile C

    is complex Gaussian with

    v

    E N(O,RU,,)

    3)

    2.2. Downlink

    If the baseband signal transmitted at the antenna airay is

    xd(t),

    hen the signal r k t ) received at the k:th mobile is

    given by

    where nk ( t ) s temporally white Gaussian noise with vari-

    ance affkand the array response vector

    vi

    corresponding to

    mobile is complex Gaussian with

    vi E

    N(O,Rt,)

    .

    In general, Rtkand R$k an be full rank matrices, even

    though in many applications several of the eigenvalues are

    significantly smaller than the noise variance, which means

    that numerically, the rank can be considered lower than the

    number of antenna elements. Examples of this kind of mod-

    els can be found in [

    1,8].

    3. ALGORITHMS

    We consider beamformers of the form 1

    ( t )

    = w*x

    t )

    or

    estimation of the signal from mobile number one. Similarly,

    we use beamformers of the form xd(t)= wsl(t) for trans-

    mission of the signal

    s l ( t )

    o the first mobile.

    3.1.

    Maximal

    SINR

    In the uplink, the signal to interference plus noise ratio is

    given by [3,17]

    Define the interference plus noise covariance matrixRIN=

    xi=,

    tk+ u i 1

    Then the maximum

    SINR

    beamformer

    is given by the eigenvector corresponding to the maximum

    eigenvalue of the generalized eigenvalue problem

    R;,W = XRINW. (7)

    It is easy to show that

    RIN

    in

    (7)

    can be replaced by

    R:

    =

    Rtl +RINwithout changing the solution. Thus, the max-

    imum SINR beamformer can alternatively be characterized

    by (up to a scaling)

    arg

    m a w*RZw (8)

    W'R w =l

    1

    which

    is

    a Quadratically Constrained Minimum Variance

    (QCMV) beamformer, closely related to the linearly con-

    strained minimum variance or Capon beamformer [10,141.

    One tempting implementation of (8) is to estimate

    Rtl

    using a training sequence or a blind DOA based method and

    use this estimate together with the unstructured estimate

    9 )

    t=l

    in the calculation of w.

    However, as will be illustrated in

    Section 4, this can easily lead to problems with signal can-

    cellation. Just as for the Capon beamformer, several meth-

    ods

    can be used to avoid this problem. One solution is to

    use a structured estimate also for RU,,as suggested i? [113,

    another is to use diagonal loading, i.e., to replace Ri by

    RU,

    PI,

    see e.g.

    [6]

    For the downlink, a design criterion similar to (7) can

    be derived as the beamformer that gives maximum signal

    power at the desired mobile while keeping the total power

    transmitted to all other users below a certain threshold. The

    details can be found

    in

    [18]. Similarly, the criterion (8) can

    motivated as the beamformer that transmits the minimum

    total power to all users while keeping the power transmitted

    to the desired user at a fixed level.

    3.2. MMSE

    A direct application of the data model (1)-(3) on the Mini-

    mum Mean Square Error (MMSE) criterion would give the

    all-zero solution, since

    E[v ]

    = 0 which results in

    E[s;(t)x (t)] = 0. However, this is only a problem with

    the mathematical treatment, since a coherent detector can

    track the phase of the signal. In the traditional plane wave

    models, this problem is handled mathematically fixing one

    element of the array response vector to 1. This procedure

    cannot be directly used on the fading channel, so we give

    two alternative solutions to the problem.

    Perform an eigenvalue decomposition of

    Rtl ,

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    Then, each realization of vy can be written as

    m

    ; + +

    0.9

    +

    +

    +

    k = l

    where

    Pk

    are independent Rayleigh distributed random vari-

    ables with E[lpk12]

    =

    x k and

    k

    are independent uniformly

    distributed over [0 ,27~] .

    Now, let

    (t)=

    e-j41xu(t), then the MMSE beam-

    former for s I C ) given ( t )s

    w = E[jis;(t)] c (R ,)-'el (13)

    The same

    w

    can be appl