A novel simplified channel tracking method for MIMO–OFDM

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    Signal Processing 88 (2008) 10021016

    A novel simplified channel tracking method for MIMOOFDM

    systems with null sub-carriers

    Hyoung-Goo Jeona, Erchin Serpedinb,

    aDepartment of Information Communication Engineering, Dongeui University, Busan, Republic of KoreabDepartment of Electrical and Computer Engineering, Texas A&M University College Station, TX 77843-3128, USA

    Received 10 April 2007; received in revised form 7 August 2007; accepted 19 October 2007

    Available online 26 November 2007

    Abstract

    This paper proposes an efficient scheme to track the time variant channel induced by multi-path Rayleigh fading in

    mobile wireless multiple input multiple outputorthogonal frequency division multiplexing (MIMOOFDM) systems with

    null sub-carriers. In the proposed method, a blind channel response predictor is designed to cope with the time variant

    channel. The proposed channel tracking scheme consists of a frequency domain estimation approach that is coupled with a

    minimum mean square error (MMSE) time domain estimation method, and does not require any matrix inverse

    calculation during each OFDM symbol. The main attributes of the proposed scheme are its reduced computational

    complexity and good tracking performance of channel variations. The simulation results show that the proposed method

    exhibits superior performance than the conventional channel tracking method [Y.G. Li, N. Seshadri, S. Ariyavisitakul,

    Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels, IEEE J. Sel. Areas

    Commun. 17 (March 1999) 461471] in time varying channel environments. At a Doppler frequency of 100 Hz and bit

    error rates (BER) of 104, signal-to-noise power ratio (Eb=N0) gains of about 2.5 dB are achieved relative to theconventional channel tracking method [Y.G. Li, N. Seshadri, S. Ariyavisitakul, Channel estimation for OFDM systems

    with transmitter diversity in mobile wireless channels, IEEE J. Sel. Areas Commun. 17 (March 1999) 461471]. At a

    Doppler frequency of 200 Hz, the performance difference between the proposed method and conventional one becomes

    much larger.

    r 2007 Elsevier B.V. All rights reserved.

    Keywords: Channel; Estimation; MIMO; OFDM; Tracking; Fading; Doppler

    1. Introduction

    The multiple input multiple output (MIMO)

    technique represents an efficient method to increase

    data transmission rate without increasing band-

    width since different data streams are transmittedfrom each transmit antenna [1]. Recently, orthogo-

    nal frequency division multiplexing (OFDM) has

    been effectively used for transmitting high speed

    data in multi-path fading channel environments. In

    OFDM, the high speed data stream is processed in

    parallel and transmitted by N (in general, a power

    of 2) orthogonal sub-carriers. The high spectral

    efficiency of OFDM and its robustness to multi-

    path fading channel environments are the main

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    0165-1684/$- see front matterr 2007 Elsevier B.V. All rights reserved.

    doi:10.1016/j.sigpro.2007.10.017

    Corresponding author. Tel.: +1 979 458 2287;

    fax: +1979 8624630.

    E-mail addresses: [email protected] (H.-G. Jeon),

    [email protected] (E. Serpedin).

    http://www.elsevier.com/locate/sigprohttp://dx.doi.org/10.1016/j.sigpro.2007.10.017mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.sigpro.2007.10.017http://www.elsevier.com/locate/sigpro
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    reasons for its widespread usage in high bit-rate

    transmissions such as digital audio broadcasting

    (DAB), digital video broadcasting (DVB) and

    wireless local area networks (WLAN) [2]. The

    combined transmission method of MIMOOFDM

    has attracted a lot of attention as a new datatransmission method in high speed data rate

    systems. In MIMOOFDM receivers, the estimated

    channel frequency response is used to separate the

    mixed signals received from multiple antennas. An

    important aspect is the fact that the performance of

    MIMOOFDM receivers highly depend on the

    accuracy of the channel estimator.

    Thus far, numerous studies for channel estima-

    tion in MIMOOFDM systems have been reported

    (see e.g., [310]). Among the most notable results, Li

    proposed a MMSE channel estimation method [3]

    that exhibits good accuracy. However, this methodis computationally very complex due to the inverse

    matrix calculation. In [6], by exploiting the correla-

    tion of the subcarrier responses, Minn et al.

    proposed a low complexity channel estimation

    method which reduced the inverse matrix size by

    half. However, Minn et al. method may cause

    channel estimation errors in large delay spread

    environments. Li also proposed a simplified channel

    estimation method which required no matrix inver-

    sion [4]. However, as mentioned in [6], if null sub-

    carriers are used, Qiin of [4] would not be theidentity matrix, and there may be some performance

    degradation according to the number of null sub-

    carriers. Since real OFDM systems have null sub-

    carriers in the guard band, a low complexity channel

    estimation method considering null sub-carriers is

    still needed.

    As a possible solution to these problems, we are

    proposing a novel simplified channel tracking

    method that relies on a blind channel predictor.

    The proposed method does not require prior

    channel information or matrix inversion calculation

    at all. In addition, the proposed method can

    effectively track the nonlinear time varying channel

    by using a piecewise linear model. To reduce its

    computational complexity while maintaining a good

    tracking accuracy, the proposed channel tracking

    scheme is built by coupling a frequency domain

    estimation approach with an MMSE time domain

    channel estimation approach.

    The remainder of this paper is organized as

    follows. In Section 2, the MIMOOFDM system

    and channel model are briefly described. Section 3

    introduces the proposed channel tracking method.

    In Section 4, the mean square error (MSE) and

    computational complexity of the proposed channel

    tracking scheme are assessed. The performance of

    the proposed method is corroborated by computer

    simulations in Section 5. Finally, Section 6 con-

    cludes the paper.

    2. Channel and MIMOOFDM system description

    The channel impulse response of the mobile

    wireless channel [3,6] can be modeled by

    ht; t XL1k0

    aktdt tk, (1)

    where akt denotes the complex gain of the kth

    path, tk represents the delay of the kth path, L is the

    number of the multi-paths in the channel and dtstands for the impulse function. The frequency

    response at time t is given by

    Ht;f9

    Z11

    ht; t ej2pft dt XL1k0

    akt ej2pftk.

    2

    Considering the motion of the mobile station, the

    path gains akts are modeled to be independent

    wide-sense stationary, narrow band complex Gaus-

    sian processes and to have different average powers

    s2k. With tolerable leakage, the channel frequencyresponse can be expressed as [3]

    Hl; k9HlTf Tg; kDf XL01n0

    hl; nWknN , (3)

    where hl; n9hlTfTg; nts, WN9 expj2p= N,L0 stands for the channel length and depends on the

    time dispersion of the wireless channel, N is the

    number of tones and the fast Fourier transforma-

    tion/inverse fast Fourier transformation (FFT/

    IFFT) size, Tf

    and Df denote the OFDM symbol

    period and sub-carrier spacing of the OFDM

    system, respectively, Tg represents the guard time

    Tg9Tf=4 and ts is the sample interval given byts 1=NDf.

    In a MIMOOFDM system, the output signal at

    each Rx (receive) antenna is a mixed signal

    consisting of the data streams coming from all Tx

    (transmit) antennas. If the channel response

    does not change during one OFDM symbol

    and the cyclic prefix is longer than the channel

    response length, the receive signal at the jth Rx

    antenna can be expressed in the frequency domain

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    as follows:

    Rjl; k XNti1

    Hijl; kXil; k Wjl; k,

    j 1; . . . ; Nr; 0pkpN 1, 4

    where Hijl; k is the channel frequency responsecorresponding to the kth sub-carrier and the lth

    OFDM symbol transmitted between the ith Tx

    antenna i 1; . . . ; Nt and the jth Rx antenna.Also, let N, Nr and Nt denote the number of sub-

    carriers, the number of Rx antennas, and the

    number of Tx antennas, respectively. Xil; k de-notes the data transmitted from the ith Tx antenna

    on the kth sub-carrier at the lth OFDM symbol.

    Wjl; k represents the additive white Gaussian noise(AWGN) at the jth receiver antenna, with zero

    mean and variance s2n, and is assumed to beuncorrelated for different js, ks, or ls. Under the

    assumption that the channel stays constant within

    one OFDM symbol duration but the channel

    changes from symbol to symbol, we will develop a

    channel tracking scheme with improved perfor-

    mance relative to the conventional scheme [3]. The

    computer simulations, which assume realistic Ray-

    leigh fading conditions (that are not limited to the

    block fading assumption) [11], corroborate the

    superior performance of the proposed channel

    tracking scheme. The indices n and k denote timeand frequency-domain indices, respectively. The

    symbols ~a, a and a denote the temporally estimated

    value, the estimated value and the predicted value of

    the variable a, respectively. In this system model,

    time synchronization is assumed to be perfect, and

    the maximum likelihood (ML) detection method is

    used. Tx antennas transmit a long preamble

    consisting of two training symbols before data

    transmission mode, as WiBro and WLAN systemsdo [2,12]. It is assumed that in the data transmission

    mode, Nd OFDM symbols are transmitted con-

    secutively in each Tx antenna. For unbiased

    performance comparison of channel tracking algo-

    rithms, no channel coding is used.

    3. Proposed channel estimation method

    Since the wireless channel is time-variant, it is

    necessary to track the channel response continu-

    ously. In addition, since the received signal at eachRx antenna in MIMOOFDM systems is a multi-

    ple-input single-output (MISO) signal, a time

    domain channel estimation cannot be directly

    applied on the received signal. In this paper, we

    propose a low complexity adaptive channel estima-

    tion method based on a blind channel prediction

    scheme that is suitable for time variant channel

    environments. The conceptual block diagram of

    the proposed channel tracking scheme is shown in

    Fig. 1. Before channel estimation, the frequency

    domain MISO signal received at the jth Rx antenna(Rjl; k) is converted into the desired single-inputsingle-output (SISO) signal (See Section 3.2 for the

    definition of desired SISO signal) by canceling the

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    FFTrj[l,n] Rj[l,k]

    Xi[l,k]

    Sij[l,k] Hij[l,k] Hij[l,k]

    hij[l,n]

    MISO

    to

    SISOconversion

    Freq.domain

    Channel

    Est.

    IFFT FFT

    Timedomain

    Channel

    Estimation

    Post ML

    detector

    SISOMISO

    Hij[l,k]Pre-ML

    detector

    Channel

    predictor

    Delay

    device

    Fig. 1. Block diagram of the proposed channel tracking method.

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    other interfering signals coming from the other Tx

    antennas. When pre-demodulating and converting

    the MISO signal Rjl; k into the desired SISOsignal, the predicted channel response Hijl; k isused to cope with the time variant channel instead

    of using the previously estimated channel responseHijl 1; k. Once the SISO signal is obtained,temporal channel estimation in frequency domain

    is carried out to remove the inverse matrix calcula-

    tion required in the following time domain channel

    estimation. In the time domain channel estimation

    block, the channel impulse response is obtained by

    minimizing a MSE cost function, considering the

    presence of null sub-carriers.

    3.1. A blind channel response predictor design

    In this paper, we design a blind channel predictor

    by exploiting a piecewise linear model for the time

    varying channel. Fig. 2 shows an example of the

    time varying channel frequency response at the kth

    sub-carrier. The channel frequency response of each

    sub-carrier, Hijl; k, varies nonlinearly with time.However, the adjacent channel frequency responses

    Hijl; k and Hijl 1; k present a certain correla-tion with each other and this relation is expressed as

    Hijl; k Hijl 1; k Dijl; k, (5)

    where Dijl; k denotes the difference between thechannel frequency responses corresponding to lth

    and l 1th symbols at the kth sub-carrier. If a

    piecewise linear model [13] is used during the short

    time of some OFDM symbol periods as shown in

    Fig. 2, the nonlinear time varying channel frequency

    response Hijl; k can be treated as a linear model.Using the piecewise linear model, let us assume

    that Hijl; k varies linearly during the time of theconsecutive MOFDM symbols. The variable Mcan

    be set according to the channel response changing

    rate. For example, a channel with a short coherence

    time will have a small M, and vice versa. In the

    piecewise linear model, the condition Dijl; k Dijl 1; k is assumed. Therefore, from (5) we inferthat Hijl; k Hijl 1; k Dijl 1; k. If we

    know^

    Hijl 1; k and^

    Dijl 1; k at the timeinstant corresponding to the lth symbol, thenthe predicted channel response Hijl; k can beobtained as

    Hijl; k9Hijl 1; k Dijl 1; k. (6)

    Referring to Fig. 2 of the piecewise linear model,

    Dijl 1; k can be expressed by using the previouslyestimated channel responses, since 2Dijl 1; k Hijl 1; k Hijl 3; k and Dijl 1; k Hijl 2; k Hijl 3; k. In the case where the

    channel response varies linearly during the Mconsecutive OFDM symbols, therefore, (6) can be

    expressed by the linear combination of the M 1

    previously estimated channel responses as follows:

    Hijl; k9XM1m1

    omHijl m; k

    XM1m1

    omHijl m; k Zijl m; k, 7

    where om is a weight value and Zijl; k9Hijl; k

    Hijl; k and denotes the random channel esti-mation error with zero mean and variance s2e (see

    Section 4.2). Let us define DHijl; k9Hijl; k Hijl; k as the channel response prediction randomerror. The weight values can be found by minimiz-

    ing the following MSE cost function:

    xl; k9EfjDHijl; kj2g

    E Hijl; k XM1m1

    omHijl m; k

    28>>>>>>:

    (15)

    where index k 0 denotes the DC component and g

    stands for the number of null sub-carriers in the

    guard band. Note that estimating the channel

    response in the frequency domain removes the need

    of calculating an inverse matrix in the next step of

    time domain channel estimation employing an

    MMSE technique. The time domain channel esti-

    mation is performed during the next step, byconsidering all the null sub-carriers used in the

    guard band. The time domain channel estimate

    hijl; n can be found by minimizing the followingMSE cost function:

    Cfhijl; ng; i 1; . . . ; Nt

    9XN1k0

    Yijl; kXL0n0

    hijl; nWnkN Zk

    2

    . 16

    hijl; n is the estimated channel impulse response

    and can be determined by solving

    qCfhijl; ng

    qhijl; n09

    1

    2

    qCfhijl; ng

    qRhijl; n0 j

    qCfhijl; ng

    qIhijl; n0

    ( )

    0, 17

    where R and I denote the real and imaginary

    parts of a complex number, respectively, and

    n0 0; 1; . . . ; L0. Direct solving (17) results in

    XN1k0

    Yijl; k XL0n0

    hijl; nWknN Zk !

    ZkWkn0N 0.

    (18)

    Define

    qn9XN1k0

    ZkWknN , (19)

    yijl; n9XN1k0

    Yijl; kZkWknN . (20)

    Then, (18) can be expressed as

    XL01n0

    hijl; nqn0 n yijl; n0 (21)

    for i 1; . . . ; Nt and n0 0; 1; . . . ; L0. Eq. (21) canbe expressed in matrix form as

    Qhijl yijl, (22)

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    where

    Q9

    q0 q1 q1 L0

    q1 q0 q2 L0

    ..

    . ...

    ...

    qL0 1 qL0 2 q0

    0BBBBB@

    1CCCCCA

    ,

    (23)

    hijl9hijl; 0; hijl; 1; . . . ; hijl; L0 1T, (24)

    yijl yijl; 0;yijl; 1; . . . ;yijl; L0 1T. (25)

    Hence, the channel impulse response can be

    estimated by

    hijl Q1yijl. (26)

    Since matrix Q is a time invariant constant matrixdetermined by the null sub-carrier pattern, the

    inverse matrix Q1 can be pre-calculated and stored

    in the memory before beginning the channel

    tracking. Therefore, no explicit calculation of Q1

    is required for every OFDM symbol. However, [3]

    and [6] require inverse matrix calculation for every

    OFDM symbol.

    4. MSE calculation and complexity comparison

    4.1. MSE calculation

    In this section, we will derive the MSE of the

    proposed channel estimation scheme. From (20),

    one infers that

    yijl; n XN1k0

    XL01m0

    hijl; mWmkN Vijl; k

    !ZkWnkN

    XL01m0

    hijl; mqn m vijn; 0pnpL0,

    27

    where vijn PN1

    k0 Vijl; kZkWnkN . Eq. (27) can

    be expressed in matrix form as

    yijl Qhijl vijl, (28)

    where

    hijl hijl; 0; hijl; 1; . . . ; hijl; L0 1T, (29)

    vijl vjl; 0; vjl; 1; . . . ; vijl; L0 1T. (30)

    From (28), the channel impulse response estimate

    corresponding to Tx antenna iand Rx antenna jcan

    be expressed as

    hijl Q1yijl hijl Q

    1vijl. (31)

    The MSE of the channel impulse response estimate

    can be given by

    MSEl9Efkhijl hijlk2g

    EfQ1vijlHQ1vijlg

    TracefQ1EfvijlvijlHgQ1Hg. 32

    A generic entry of EfvijlvijlHg is given by

    Efvijl; n1vijl; n2g E

    XN1k10

    Vijl; k1Zk1Wn1k1N

    !(

    XN1

    k10

    Vijl; k2Zk2Wn2k2N

    !)

    XN1

    k1;k20

    EfVijl; k1Vijl; k2g

    Zk1Zk2Wn1k1n2k2N , 33

    where n1; n2 1; 2; . . . ; L0. If we substitute (14) into(33), Eq. (33) can be rewritten as

    Efvijl; n1vijl; n2g

    XN1k0

    PNtm1;maijXml; kj

    2

    jXil; kj2

    s2p

    (

    s2n

    jXil; kj2

    )ZkW

    n1n2kN , 34

    where s2p9EjDHijl; kj2 s2e

    PM1m1 o

    2m and DHij

    l; k and Wjl; k are assumed to be independent ofeach other. Hence, if a constant modulus modula-

    tion is used,

    EfvijlvijlHg Nt 1s

    2p

    s2njXil; kj

    2

    Q: 35

    If no null sub-carriers are used, since Q NI,

    MSEl L0N

    Nt 1s2

    p 1

    SNR

    . (36)

    Given a SNR, the MSEl is dependent on the

    prediction error, the channel response length and

    the number of Tx antennas.

    4.2. Mean and variance of random variable Zijl; k

    For convenient comprehension, let us assume

    that BPSK is used for Xil; k, then EXil; k 0.Xml; k, DHijl; k, Wjl; k and Xil; k are indepen-

    dent of each other. It is clear that EXml; k 0,

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    EXil; k 0 and EWjl; k 0. Therefore, it isalso clear that EVijl; k 0, where Vijl; k is givenby (14). From (31), the channel estimation error in

    time domain can be expressed as Ehijl hijl

    Q1Evijl. Since Evijn PN1k0 EVijl; kZk

    W

    nk

    N 0, Q

    1

    Evijl 0. It means that the meanof the channel estimation error in time domain is

    zero. Since the channel frequency response estimate

    Hijl; k is obtained by performing Fourier trans-formation for the channel impulse response estimate

    hijl; n, the mean of the channel estimation error infrequency domain is also zero. That is, EZijl; k EHijl; k Hijl; k 0.

    The MSE of the channel frequency response in

    the kth sub-carrier is defined by EHijl; kHijl; k

    2 s2e . The total MSE of the channel

    impulse response in time domain is equal to

    MSEl given by (36). When the channel impulseresponse hijl; n is Fourier transformed to obtainthe channel frequency response Hijl; k, the totalMSE in time domain is equal to the total MSE in

    frequency domain. Therefore, if there are N sub-

    carriers in the OFDM system, then s2e MSEl=N.

    4.3. Computational complexity comparison

    In this section, the computational complexities of

    the schemes proposed in [3,6] and the method

    proposed herein are compared briefly. The complex-ity comparison will be focused on the channel

    estimation based on the decision-directed estimation

    method, as [6] did. Since Q1 can be pre-calculated,

    Q1 calculation is not included in the complexity

    comparison. Since Lis simplified method [4] may

    cause estimation error in MIMOOFDM systems

    with null sub-carrier and employing non-constant

    modulus modulations, Lis simplified method will

    not be discussed in this comparison. In the case of

    two Tx and Rx antennas and N sub-carriers, the

    channel estimation complexity for each method is

    given in Table 1 (refer also to [6]). In Table 1, FFTN

    denotes the number of multiplications required forthe FFT operation with size N. invL0 L0 stands

    for the number of multiplications required for L0

    L0 matrix inversion. Nu denotes the number of the

    sub-carriers used. When Nu N, no null sub-

    carrier is used. The number of FFT operations for

    Li method [3] can be easily obtained by referring to

    Fig. 3 in [3]. Considering Eqs. (15) and (16) in [3],

    the number of multiplications required for calculat-

    ing hijl can be derived straightforwardly. The

    calculation amount for Minn method [6] can be

    obtained by considering the similarity with Li

    method. Note that when Nu N and constantmodulation is used, Qiin is a identity matrix. In

    that case, Lis method needs calculating only

    invL0 L0 instead of calculating inv2L0 2L0.

    From Table 1, we can see that the proposed method

    has the lowest complexity among these methods,

    regardless of the non-constant modulation and the

    presence of null sub-carriers.

    5. Performance evaluation

    Computer simulations are carried out to evaluatethe performance of the proposed method. Two Tx

    and two Rx antennas are used for the MI-

    MOOFDM system. There are a total of 128 sub-

    carriers so that the FFT/IFFT size is 128. The DC

    component sub-carrier is not used, and 10 and 9

    sub-carriers on each end of the spectral band,

    respectively, are used as guard band. The rest of 108

    ARTICLE IN PRESS

    Table 1

    Evaluation of computational complexity for each method

    Condition Method No. of complex multiplications and divisions

    Constant modulus with Nu N Ref. [3] 3N 2L02 5FFTN invL0 L0

    Ref. [6] 4:5N 2L02 3FFTN=2 invL0 L0

    Proposed 3N 2L02 5FFTN

    Constant modulus with NuaN Ref. [3] 3N 2L02 5FFTN inv2L0 2L0

    Ref. [6] 4:5N 2L02 3FFTN=2 invL0 L0

    Proposed 3N 2L02 5FFTN

    Non-constant modulus Ref. [3] 5N 2L02 5FFTN inv2L0 2L0

    Ref. [6] 6N 2L02 4FFTN=2 2 invL0 L0

    Proposed 3N 2L0

    2 5FFTN

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    sub-carriers are used to transmit data. The OFDM

    symbol rate is 10 Ksps, and the symbol period is

    100ms, including the guard time of 20ms. The

    channel length L0 is assumed to be 18. Modulation

    in sub-carriers is QPSK. The carrier frequency is

    2.4 GHz. The multi-path Rayleigh fading channel

    assumes two rays with equal gain, and each ray has

    six multi-path delay taps. Each signal path is

    assumed to undergo an independent Rayleigh

    fading. The rms delay spread is 1:82ms. We used

    the Rayleigh fading channel simulator (Jakessinusoid sum method) openly published in

    Ref. [11]. The Doppler frequencies 40, 100 and

    200 Hz are used to represent different mobile

    environments. After completion of channel estima-

    tion by using the training signal, the system state is in

    data transmission mode. In data transmission mode,

    channel tracking for 20 consecutive OFDM symbols

    is carried out continuously, using a decision directed

    method in which the demodulated data is used as the

    reference data. The performance of the system is

    measured by the estimators MSE and bit error rates

    (BER), each averaged over 100,000 OFDM blocks.

    For unbiased comparison, no channel coding is used

    in this simulation. In order to track the time varying

    channel response, Kalman filter method may be used

    for MIMOOFDM systems as Komninakis did [14].

    However, the Komninakis method should calculate

    the inverse matrix in every OFDM symbol to obtain

    Kalman gain matrix and thereby the calculation

    amount increases significantly. For this reason, we

    compare the proposed method with Kalman filter

    estimator, using scalar Kalman filter in each sub-

    carrier [15].

    Fig. 3 shows an example of the proposed channel

    tracking and the nonlinear time variant channel

    frequency response H11l; k simulated at thegiven multi-path channel parameters, l 1; 2; . . . ;100, k 10, and maximum Doppler frequency

    fd 200 Hz. In Fig. 3, solid line is the channel

    response tracked by the proposed method at

    Eb=N0 15 dB. Fig. 3 shows that the nonlinearchannel response is well tracked by the proposed

    method. The performance simulation results are

    shown in Figs. 411. Fig. 4 shows the MSE of theproposed method at the conditions ofM 4, 5 and 6,

    and at the fixed SNR ofEb=N0 25 dB. The range ofthe normalized Doppler frequency (fdTs) is given

    from 0 to 0:03. In the case of fdTso0:02, theperformance of the proposed scheme (M 4) is the

    worst among all the investigated channel tracking

    methods. The reason is that in decision directed mode,

    the performance of the proposed channel response

    estimator is very highly affected by the prediction

    error caused by the first channel prediction just after

    two training OFDM symbols. The prediction error

    causes the demodulation error which results in MSE

    performance degradation and propagates into the

    next channel estimation. As mentioned before, when

    two training OFDM symbols are received, only two

    channel responses Hij1; k and Hij2; k areobtained, and Hij3; k; . . . ; Hij1 M; k are as-sumed to be equal to Hij2; k. Therefore, from (10),the MSEs for the first channel prediction are

    proportional to xl; k / 1:8s2e when M 4, xl; k /s2e when M 5 and xl; k / 0:68s

    2e when M 6,

    respectively. There is a larger difference in MSE

    between M 4 and 5 than between M 5 and 6.

    ARTICLE IN PRESS

    0 10 20 30 40 50 60 70 80 90 1001

    0.5

    0

    0.5

    1

    1.5

    2

    OFDM symbol index

    Real{H11(l,10)}

    Eb/No= 15 dB

    fd= 200 HzEb/No= 15 dB

    fd= 200 Hz

    0 10 20 30 40 50 60 70 80 90 1001.5

    1

    0.5

    0

    0.5

    1

    OFDM symbol index

    Image{H11(l,1

    0)}

    Channel response

    Estimated channel response

    Channel response

    Estimated channel response

    Fig. 3. An example of channel frequency response Hijl; k and tracking at fd 200Hz, k 10 and Eb=N0 15dB.

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    The noise effect in channel prediction can be

    reduced by increasing M, as shown in Fig. 4.

    However, increasing M beyond a certain limit may

    result in performance degradation due to the

    increased sensitivity to channel time variations.

    Note that MSEM 6 is larger than MSEM

    5 at fdTs40:015. We can see that when M 5, the

    proposed method shows the best performance in

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    0 0.005 0.01 0.015 0.02 0.025 0.03104

    103

    102

    101

    fd*T

    s

    MSE

    Proposed(M=4)

    Proposed(M=5)

    Proposed(M=6)

    Li original

    Fig. 4. MSE performance as a function of fdTs at a given M.

    0 0.005 0.01 0.015 0.02 0.025 0.03104

    102

    103

    100

    101

    101

    fd*T

    s

    MSE

    Eb

    /No

    = 25 dB

    Proposed CE

    No predict

    FDE

    Li original

    Fig. 5. MSE performance as a function of fdTs for each channel tracking method.

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    Fig. 4. Hereafter, M is set to 5 in all the simulations

    for performance evaluation. When M 5 and the

    OFDM symbol period is 100 ms, the time duration

    for which a piecewise linear model is assumed is

    500ms. Fig. 5 shows the MSE for each method as a

    function offdTs at the conditions: Eb=N0 25 andM 5. As we can see from Fig. 5, the MSE of the

    proposed method increases more slowly than the

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    0 5 10 15 20 25 30

    104

    101

    102

    103

    101

    100

    102

    Eb/No[dB]

    MSE

    Proposed CE

    No predict

    FDE

    FED + Km filter

    Li original

    Fig. 6. MSE performance at fd 40Hz.

    0 5 10 15 20 25 30

    107

    104

    105

    106

    102

    101

    103

    100

    Eb/No[dB]

    BER

    Proposed CE

    Perfect CE

    No predict

    FDE

    FED + Km filter

    Li original

    Fig. 7. BER performance at fd 40Hz.

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    other methods as fdTs increases. On the other hand,

    MSE of Lis original method increases rapidly when

    fdTs40:015.

    Figs. 6, 8 and 10 show MSE performances at

    Doppler frequencies of 40, 100 and 200 Hz, respec-

    tively. Figs. 7, 9 and 11 show BER performances at

    ARTICLE IN PRESS

    0 5 10 15 20 25 30

    104

    101

    102

    103

    101

    102

    100

    103

    Eb/No[dB]

    MSE

    Proposed CE

    No predict

    FDE

    FED + Km filter

    Li original

    Fig. 8. MSE performance at fd 100Hz.

    0 5 10 15 20 25 30

    104

    105

    106

    102

    101

    103

    100

    Eb/No[dB]

    BER

    FED + Km filter

    Proposed CE

    Perfect CE

    No predictFDE

    Li original

    Fig. 9. BER performance at fd 100Hz.

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    Doppler frequencies of 40, 100 and 200 Hz, respec-

    tively. In these figures, BER performance curves for

    perfect channel estimation are given to show the

    performance in the ideal channel estimation case. In

    these figures, FDE denotes the frequency domain

    channel tracking method of Ref. [10]. In order to

    compare the effect of channel prediction, BER and

    MSE curves for the no predict tracking method are

    ARTICLE IN PRESS

    0 5 10 15 20 25 30

    104

    101

    102

    103

    101

    100

    102

    Eb/No[dB]

    MSE

    Proposed CE

    No predict

    FDE

    FED + Km filter

    Li original

    Fig. 10. MSE performance at fd 200Hz.

    0 5 10 15 20 25 30

    106

    104

    10

    5

    102

    101

    103

    100

    Eb/No [dB]

    BER

    Proposed CE

    Perfect CE

    No predictFDEFED + Km filterLi original

    Fig. 11. BER performance at fd 200Hz.

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    also drawn in these figures. No predict means the

    proposed method without the channel prediction

    function. If no prediction is used, the previously

    estimated channel value Hijl 1; k is used asthe current channel value Hijl; k. In this case, the

    channel estimation error is given by Hijl; kHijl 1; k Dijl; k Zijl; k. When the channelprediction is used, the channel estimation error is

    given by DHijl; k Hijl; k Hijl; k. The MSEof no prediction is given by EHijl; kHijl 1; k

    2 VarDijl; k s2e . The MSE of

    prediction is EjDHijl; kj2 1:5s2e as we can see

    from (10). Therefore, the performance of predic-

    tion is better than that of no prediction as long as

    VarDijl; k40:5s2e is satisfied. VarDijl; k % 0 at

    a low Doppler frequency ands2e is inversely propor-

    tional to the Eb=N0 such that s2e % 0 at a high

    Eb=N0. Therefore, the lower the Doppler frequencyis in the wireless channel, the higher Eb=N0 isrequired to satisfy the condition of VarDijl; k40:5s2e . That is the reason why the performance ofno prediction is better than that of prediction at

    fd 40 Hz. Note that in Fig. 6, the performance

    gap in Eb=N0 between no prediction and predic-tion is getting narrow with increase of Eb=N0. If wecan know the information about Eb=N0 and theDoppler frequency, either no prediction or pre-

    diction can be selected to improve the performance

    of channel estimator, based on such information.At a Doppler frequency of 40 Hz, there is a little

    difference in the MSE and BER performance

    between the proposed method and Lis original

    method. At fd 100 Hz and BER of 104, the

    performance improvement provided by the pro-

    posed method is about 2.5 dB in Eb=N0 comparedwith that of Lis original method. However, at a

    Doppler frequency of 200 Hz, the BER performance

    difference becomes much larger when compared

    with those of other methods. As we can see from

    Figs. 9 and 11, due to the inter sub-carrier

    interference (ICI) [16], BER performance at a

    Doppler frequency of 200 Hz is worse than that of

    100 Hz.

    From these figures, we can observe that the

    channel estimation error of Lis original method is

    very large at a given low Eb=N0. At a given lowEb=N0, BER is high and the demodulated dataXl; k is erroneous. Since Lis original methodcalculates the inverse of a matrix made of erroneous

    demodulated data Xl; k, the channel estimationerror is amplified by noise and the erroneous inverse

    matrix. In the worst case of low Eb=N0 (less than

    5 dB), the channel estimation error may diverge when

    tracking the channel response. On the other hand,

    since the proposed method uses the known inverse

    matrix Q1 which is a constant time-invariant

    matrix, the effect of the erroneous demodulated data

    is much less significant than that of Lis originalmethod. The simulation results show that as expected

    the proposed method does not diverge.

    6. Conclusions

    This paper proposed a novel simplified channel

    tracking method to reduce the computational

    complexity and improve the tracking performance

    in time varying channel environments. In the

    proposed method, a blind channel response pre-

    dictor is designed to cope with the time variant

    channel. The proposed channel tracking scheme

    consists of a frequency domain estimation approach

    that is coupled with an MMSE time domain

    estimation method, and does not require any matrix

    inverse calculation during each OFDM symbol. By

    converting the MISO signal into a SISO signal and

    performing temporal channel estimation in the

    frequency domain before beginning time domain

    channel estimation, no matrix inversion is required

    anymore. The simulation results show that the

    proposed method exhibits superior performance

    than Lis original method in time varying channelenvironments. At a Doppler frequency of 100 Hz

    and BER of 104, signal-to-noise power ratio

    (Eb=N0) gains of about 2.5 dB are achieved relativeto Lis original method. At a Doppler frequency of

    200 Hz, the performance difference between the

    proposed method and conventional one becomes

    much larger.

    Acknowledgment

    This work was partially supported by researchproject 07-03 funded by Electronics and Telecom-

    munications Research Institute (ETRI) in Korea.

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