Improved Channel Estimation for Slow to Moderate Fading Channel in OFDM Systems using Random Walk Based Kalman Filter

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Improved Channel Estimation for Slow to Moderate Fading Channel in OFDM Systems using Random Walk Based Kalman Filter

K. Vidhya

ECE Department, JCET Trichy, India

M. Geetha

HOD, ECE Department, JCET Trichy, India

Abstract This project deals with estimating the multi path channel condition in orthogonal frequency division multiplexing system. Many estimation algorithms exploit the time-domain correlation of the channel by employing a Kalman filter based on an approximation of the time-varying channel. Least square estimator explores the frequency domain correlation of the channel and the knowledge of the delays to convert the pilot frequencies into a primary estimate of the path Complex Amplitude. It also defines an error signal for each path. The lower dimensional RW-KF that estimates the complex amplitude of each path separately from the LS estimated signal. We demonstrate that this amounts to a simplification of the joint multi-path Kalman gain formulation through the Woodburys identities. Hence, this new algorithm consists of a superposition of independent single-path single-carrier KFs. This observation allows us to adapt the optimization to the actual multi-path multi- carrier scenario, to provide analytic formulas for the mean- square error performance and the optimal tuning of the proposed estimator directly as a function of the physical parameters of the channel (Doppler frequency, signal-to-noise-ratio). These analytic formulae are given for the first-, second-, and third-order RW models used in the KF. The proposed per-path KF is shown to be as efficient as the exact KF (i.e., the joint multi-path KF).

KeywordsOrthogonal frequency division multiplexing; Channel estimation; Least square estimator; Random walk model; Kalman filter.

  1. INTRODUCTION

    Orthogonal frequency division multiplexing (OFDM) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, whether wireless or over copper wires, used in applications such as digital television and audio broadcasting, DSL Internet access, wireless networks, power line networks, and 4G mobile communications. OFDM is a frequency-division multiplexing (FDM) scheme used as a digital multi-carrier modulation method. A large number of closely spaced orthogonal sub- carrier signals are used to carry data on several parallel data streams or channels. Each sub-carrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation or phase-shift keying) at a low, symbol rate maintaining total data rates similar to the conventional Single-carrier modulation schemes in the same bandwidth.

    The primary advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions (for example, attenuation of high frequencies in a long copper wire, narrowband interference and frequency-selective fading due to multipath) without complex equalization filters. Channel equalization is simplified because OFDM may be viewed as using many slowly modulated narrowband signals rather than one rapidly modulated wideband signal. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate inter symbol interference (ISI) and utilize echoes and time- spreading (on analogue TV these are visible as ghosting and blurring, respectively) to achieve a diversity gain, i.e. a signal-to-noise ratio improvement. This mechanism also facilitates the design of single frequency networks (SFNs), where several adjacent transmitters send the same signal simultaneously at the same frequency, as the signals from multiple distant transmitters may be combined constructively, rather than interfering as would typically occur in a traditional single-carrier system.

    Orthogonal Frequency-Division Multiplexing (OFDM) is an effective technique for alleviating frequency-selective channel effects in wireless communication systems. In this technique, a wideband frequency-selective channel is converted to a number of parallel narrow-band flat fading sub channels which are free of Inter-Symbol-Interference (ISI) and free of Inter-Carrier Interference (ICI) (for negligible channel time variation within one OFDM symbol period T). For coherent detection of the information symbols, reliable estimation of the channel in the OFDM system is crucial.

  2. PROBLEM FORMULATION

    Most of the conventional methods work in a symbol-by symbol scheme [1][3] using the correlation of the channel only in the frequency domain; i.e., the correlation between the sub channels. More advanced algorithms are based on the Kalman Filter (KF), to also exploit the time-domain correlation. This paper deals with channel multi-path Complex Amplitude (CA) estimators based on KFs. An approximation often used in the literature consists of approaching the fading process as auto-regressive [4]. Hence,

    a widely used channel approximation is based on a first-order Auto-Regressive model (AR1), as recommended by [4], combined with a Correlation- Matching (CM) criterion to fix the AR1 coefficient. The KF channel estimator that results from this choice, hereafter called AR1CM -KF, has been used in several studies concerning various systems, such as in multiple-input-multiple- output systems [4], and in OFDM systems [4][6]. The AR1CM -KF appears to be convenient for the very high mobility case, which leads to quasi-optimal channel estimation performance compared to lower bounds, as seen, for example, in [5], [6] (in these studies, the AR1 – KF is actually used to track the basis extension model coefficients of the high-speed channel). However, here we consider moderate normalized Doppler frequency (fdT)

    values; i.e., fdT10-2. This corresponds to low mobility ( 50

    km/h) with the actual systems such as Worldwide Interoperability for Microwave Access (WiMAX) Mobiles. However, with the development of the cognitive radio, lower carrier frequencies are investigated for future systems.

    For instance, VHF/UHF television broadcast bands from 54 MHz to 862 MHz [8] and aeronautical bands from 960 MHz to 1215 MHz are planned to be deployed. For a given fdT, as the speed is inversely proportional to the carrier frequency, fdT values around 10-2 can correspond to a relative high mobility with such systems (hundreds of km/h). This prompts the need for a comprehensive study of channel estimation for fdT10 -2.

    For this scenario, whereby the channel variation within one symbol duration can be neglected ([3][5], [7]), the AR1CM -KF estimator usually exploited in the literature is far from being effective [7]. A better tuning of the AR1 coefficient can focus on minimizing the estimation variance in the output of the KF, as proposed by [8] (with the analytic mean-square error (MSE) performance for a given Doppler and signal-to-noise ratio (SNR) scenario. This performance can be obtained by a first-order Random Walk (RW)-model based KF (RW1-KF) ([8]).

    On the other hand, it has been shown recently that the MSE performance of a KF can still be improved by switching from the AR1 model to an integrated RW model (also called

    pilot subcarriers into L multi-path values where L is the number of multi-paths and applies a KF to each path.

  3. SYSTEM DESCRIPTION

    1. OFDM System Model

      Orthogonal frequency division multiplexing (OFDM) is a parallel transmission scheme, where a high-rate serial data stream is split up into a set of low-rate sub streams, each of which is modulated on a separate sub-carrier (SC) (frequency division multiplexing). Let us consder an OFDM system with N subcarriers, and cyclic prefix length Ng. The duration of an OFDM symbol is T = NTTS, where TS is the sampling time and NT = N+Ng. Let X(k) is the sequence of transmitted elementary symbols of the k-th OFDM symbol. X(k) transmitted symbol is modulated by (M-PSK). Modulate symbol transmitted on the subcarrier with indice n-1-N/2. The sequence of transmitted symbol is assumed to be zero mean and stationary with normalized variance. After transmission over a slowly time-varying multi-path channel and fast Fourier transform demodulation, the k-th received OFDM symbol Y(k) is given by

      Y(k) = H(k) X(k) + W(k) (1)

      ) e

      )

      ) e

      )

      x.

      x.

      )

      )

      Where W(k) is a complex circular Gaussian noise vector and H(k) is a diagonal matri

      ) is

      )

      ) is

      )

      2)

      2)

      (

      (

      [H(k)] = 1/N ( ( (

      (

      (

      Where L is the total number of propagation paths, ( a CA of the l-th path at k-th OFDM symbol.

    2. Pilot Pattern

      We use Np pilot sub-carriers, they are evenly inserted into the N sub-carriers. The received pilot sub-carriers can be written as

      Yp(k) = diag{Xp(k)}Fp (k) + Wp(k) (3)

      ub

      ub

      -c

      -c

      a

      (

      a

      (

      riers

      )

      riers

      )

      ,

      ,

      Where Xp, Yp and Wp which correspond to the sent and received data symbol, and the channel noise on the pilot sub- carriers, respectively. The Np×L matrix Fp is the Fourier

      the integrated Brownian model) for the approximation model.

      A second-order RW model and a third-order RW model have

      matrix of the pilot s

      r with elements given by

      been respectively considered in [8] and [9]. They take into account that the exact path CA continues in a given direction during several symbols for low fdT, and shows strong trend behaviour. The Kalman estimators based on these second order and third order models are here called the RW2-KF and RW3-KF estimators, respectively. The RW-KF estimators of the previously cited studies were designed for single-path

      [Fp]np,l = e (4)

    3. Least Square Estimator

      To define an error signal for each path, we use the least- square (LS) estimator. The error signal for each path can be reduced by RLS filter and also it can estimate the path CAs

      obtai

      ^ (k)

      obtai

      ^ (k)

      b

      b

      channel estimation in single-carrier systems. In the present study, we consider multi-path channel estimation in multi

      obtained from the estimate is ned

      current OFDM y

      symbol Yp(k). The LS

      carrier systems (i.e., OFDM systems). In this context, we are interested in devising simplified methods compared to the

      = (F F )

      ) (5)

      F Yp(k

      high-dimensional KFs that perform joint estimation of the path CAs. Some simplified methods have lately been proposed in [10][12]. Reference [12] converts the vector of

    4. Single Path RW-KF

      We use the LS estimation of instead of Yp(k) as the input signal to reformulate the KF, and impose independent

      ^ (k), the l

      ^ (k), the l

      r as th

      . Th path i

      r as th

      . Th path i

      essing

      ^( )(k)

      Also,

      essing

      ^( )(k)

      Also,

      mo

      ^( )(k)

      mo

      ^( )(k)

      the

      ) +

      )

      the

      ) +

      )

      pat

      ( )(k)

      pat

      ( )(k)

      proc of the L paths. The l-th element of denoted by , corresponds to the LS estimation of th path CA. let us define the LS estimation erro e loop noise applied on the per-path KF, denoted by w en, the state-space del of per- h KF for the l-th s given

      ath

      ath

      ^ (

      ath

      ath

      ^ (

    5. Joint Multi-P

      mates

      / )

      mates

      / )

      =

      =

      Joint multi-p CA to esti

      RW-KF

      RW

      / )

      RW

      / )

      by

      by

      u

      )

      u

      )

      Wood

      ( )

      Wood

      ( )

      b

      b

      ositio ury

      ( )

      ositio ury

      ( )

      -KF super p ning the per-path sing s identity.

      ( )(k)

      ( )(k)

      by = ( w

      (6)

      ^ ^ /

      + K . (10)

      (

      F ) F Wp(

      Where,

      ( (

      )

      )

      /

      /

      )

      )

      = M ^

      ^

      ace n b

      ( ) .

      ace n b

      ( ) .

      e

      T

      e

      T

      he sta

      ^( )(k)

      he sta

      ^( )(k)

      here

      here

      W , w = (F k) (7)

      T te-sp model of the per-path KF for the l-th path CA ca applied to single-path RW-KF which

      ( (

      /

      )

      S

      /

      )

      S

      =

      (SP

      =

      (SP

      ( (

      ( (

      P^ M P^

      ( )=

      ( )=

      P

      P

      S

      S

      ( )

      ( )

      /

      (11)

      +

      M

      +

      M

      ( )

      ( )

      ( )

      ( )

      (

      (

      U 12)

      /

      /

      )

      )

      estimates ^ / ) he state model of the L-path CAs can be

      K iag{ } 13)

      (n vecto

      -1)+u

      ( )

      ( )

      -1)+u

      ( )

      ( )

      ( )

      ( ) (

      ( )

      ( ) (

      expressed i r form as

      (

      (

      )

      )

      ( )

      ( )

      )

      ( )

      ( )

      )

      a(k) = Ma(k k) (8)

      + d ) (

      ^ (k)

      ^ (k)

      )

      )

      =

      =

      ( /

      a

      a

      Where, a(k) = [ . ] ith s the

      state

      ) as

      state

      ) as

      ( (

      ^ (14)

      vector of the l-th path, u(k) = [u( w . u( ] with u(

      the state noise vector of the l-t th, IL×M i

      channel state evolution matrix.

      ) )

      h pa M =

      )

      s the

      ( )

      ( )

      ( )

      ( )

      ^ (

      ^ (

      ( )

      )

      ( )

      )

      ^ (

      ^ (

      ( )

      ) (

      ( )

      ) (

      ( )

      ( )

      ( )

      ( )

      TABLE I. List of Terms in the KF State Space Model

      ix S a(k) in th ( ) ,

      ( )

      tion

      ix S a(k) in th ( ) ,

      ( )

      tion

      e

      u

      m

      e

      u

      m

      t

      t

      quat

      ) , a

      )

      atrix

      quat

      ) , a

      )

      atrix

      (k

      (k

      )

      )

      Variables

      RW1

      RW2

      RW3

      )

      [ ^( ) ( ) ]

      ( ) ( )

      [ ( ) ] ]

      )

      u(

      (

      [0

      [0 0 ( ) ]

      ( )

      M

      1

      [1 1] [1 1 ½]
      [0 1] [0 1 1]
      [0 0 1]

      S

      1

      [1 0] [1 0 0]

      U(

      2 (l) u

      [0 0] [0 2 (l)]

      u

      [0 0 0] [0 0 0] [0 0 2 (l)]

      Variables

      RW1

      RW2

      RW3

      )

      [ ^( ) ( ) ]

      ( ) ( )

      [ ( ) ] ]

      )

      u(

      (

      [0

      [0 0 ( ) ]

      ( )

      M

      1

      [1 1] [1 1 ½]
      [0 1] [0 1 1]
      [0 0 1]

      S

      1

      [1 0] [1 0 0]

      U(

      2 (l) u

      [0 0] [0 2 (l)

      u

      [0 0 0] [0 0 0] [0 0 2 (l)]

      We define the multi path selection matr = IL×S. This matrix allows us to pass from the vector o ) using (k) = Sa(k). By defining FS = FPS, we obta e ion of

      (

      ( )

      ( )

      ( )

      ( )

      ( )

      ( )

      ( ) ]

      ( ) ]

      Fig. 1. Per-path KF structure.

      This CA is compared with the CA of OFDM transmitted symbol X(k). If the CA of joint multi path RW-KF is less than the CA of OFDM transmitted symbol X(k), then we can know that the channel is slow to moderate fading channel. If the CA of joint multi path RW-KF is higher than the CA of OFDM transmitted symbol X(k), then we can know that the channel is fast fading channel. If the CA of joint multi path RW-KF is equal to the CA of OFDM transmitted symbol

      the joint multi-path KF. The path variables ( nd the X(k), then we can know that there is no fading in multipath

      path

      path

      KF

      /

      KF

      /

      -th

      -th

      be w

      /

      be w

      /

      path state evolution matrix M, path selec S are defined in following table according to the (model order. The

      channel.

      ( )

      / )

      ( )

      / )

      single

      ^ (

      Where,

      RW- for the l p

      =

      =

      )

      )

      )

      )

      )

      )

      )

      )

      /

      /

      ) (

      )

      ) (

      )

      ath can

      ^( )

      ( )

      ath can

      ^( )

      ( )

      )

      /

      )

      /

      )

      )

      (

      (

      (

      (

      ^( + K(

      )

      )

      )

      )

      /

      /

      ( )

      /

      ( )

      /

      )

      )

      (

      (

      (

      (

      (

      (

      ^( = M ^

      thus

      S ^(

      ritten as

      )

      )

      )

      )

      ) (9)

      (10)

      Slow fading arises when the coherence time of the channel is large relative to the delay constraint of the channel. In this regime, the amplitude and phase change imposed by the channel can be considered roughly constant over the period of use. Slow fading can be caused by events such as shadowing, where a large obstruction such as a hill or large building obscures the main signal path between the

      ) =

      )

      ) =

      )

      (

      (

      M P(

      )

      M P(

      )

      =

      )

      /

      =

      )

      /

      M

      )

      M

      )

      +

      +

      )

      /

      )

      /

      U

      U

      P

      P

      ^ ^ (

      (

      transmitter and the receiver. The received power change

      )

      )

      ( )

      ( )

      (11) caused by shadowing is often modeled using a log-normal

      K( P(

      S /S P(

      S + (12) distribution with a standard deviation according to the log-

      ( ( (

      distance path loss model.

    6. Asymptotic Mean Square Error of the Per-Path KF

    arts, o aramet

    . Th

    arts, o aramet

    . Th

    f

    (

    f

    (

    ve

    (

    ve

    (

    l, w

    (

    l, w

    (

    The variance of estimation error is then comprised of two p ne of which comes from the variation of the p er , and the other comes from the input loop noise

    or a

    ) =

    or a

    ) =

    n p

    ) +

    n p

    ) +

    (

    w

    (

    w

    w us gi ath

    ) r

    ) r

    e have

    )

    (15)

    nt

    ) ,

    )

    mpon

    nt

    ) ,

    )

    mpon

    sult can

    (

    sult can

    (

    The compone e s from the high pass filtering

    (

    (

    of the input CA (

    hich

    be expressed in the frequency

    domain. The co ent ) r esults from the low pass

    n is

    )

    n is

    )

    calcul

    calcul

    io

    (

    io

    (

    m

    m

    a

    a

    e channe

    = 1

    e channe

    = 1

    filtering of the input loop noise. T he global mean MSE (per path) of th l esti t ated by

    /L (16)

    The integrated RW-KF allows us to provide analytic formulas for the mean square error performance. The MSE performance can still be improved by proposed estimator.

  4. SIMULATIONS

    In this simulation section, The PSK-OFDM system with N=128 sub-carriers used to validate the proposed approximate method and the analytic results. By default, the OFDM system has Ng=0.5 samples and Np=16 pilot sub- carriers in each transmitted OFDM symbol. After that this OFDM symbol is applied to RLS filter and LS estimator to define the error signal and to calculate the CA. Then, it can be applied to Single Path RW-KF which estimates CA of each path separately. Finally each separate path CAs are super positioned to get multi path CA which is compared with OFDM symbol CA to estimate the channel condition. And also the MSE performance of per path and joint multi path RW-KF, MSE of Per Path KF Versus SNR, MSE comparison between Per Path and Joint Multi-path RW-KF are shown.

    Fig. 2. Input signal.

    Fig. 3. Hk signal.

    Fig. 4. Yk signal.

    Fig. 5. Fp signal.

    Fig. 6. Ypk signal.

    Fig. 7. RLS Filtered signal.

    Fig. 8. Single path RW-KF signal.

    Fig. 2 shows the input (Xk) signal with 128 sub-carriers and cyclic prefix length is 0.5 samples. Fig. 3 is the carrier signal for PSK modulation. Fig. 4 shows the OFDM symbol with

    128 sub-carriers which is generated by input signal modulated with carrier signal and summed with white Gaussian noise signal. Fig. 5 represents the Fourier matrix of the pilot sub-carriers. Fig. 6 presents the OFDM symbol with pilot sub-carriers which is generated by the Fourier matrix and diagonal elements of input signal with pilot sub-carriers.

    Fig. 9. Joint Multi path RW-KF signal.

    Fig. 10. MSE Of Joint Multi Path RW-KF And Per Path RW-KF.

    Fig. 11. MSE Of Per Path KF Versus SNR.

    Fig. 7 shows the RLS filtered signal which can minimize the error signal in OFDM symbol. Fig. 8 and Fig. 9 shows the output of single-path RW-KF and Joint multi-path RW-KF respectively. This joint multi-path CA is comparable with the input signal CA to analyze the Multi-path channel condition. Fig. 10 presents MSE of Joint Multi Path RW-KF and Per Path RW-KF. Fig. 11 shows MSE of Per Path KF versus SNR.

    Fig. 12. MSE versus fdT

    Fig. 13. MSE Comparison Between Per Path and Joint Multi Path RW-KF

    Fig. 12 shows the MSE versus fdT and Fig. 13 shows MSE comparison between Per Path and Joint Multi Path RW-KF.

  5. CONCLUSION

This paper proposed the integrated RW-KF for channel condition estimation. Our solution is a two-step solution: first, an error signal and CA for each channel path is calculated with the LS criterion. Secondly, based on this error signal, a KF is applied to each path independently. This per path KF solution explores the time-domain correlation of the channel, while the LS step exploits the frequency-domain correlation of the channel. We have shown how to apply the previous results we obtained for a single-path single-carrier to the multi-path multi-carrier context. Lower dimensional RW-KF that estimates the complex amplitude of each path separately. Furthermore, we have demonstrated that our per- path KF solution can be interpreted as a simplified version of the more complex joint multi-path KF. This has been done through the Woodburys identities. Hence, this new algorithm consists of a superposition of independent single-path single- carrier KFs, which were allows us to adapt the optimization to the actual multi-path multi-carrier scenario, to provide channel condition estimation analytic formulas for the mean- square error performance andthe optimal tuning of the proposed estimator directly as a function of the physical

parameters of the channel (Doppler frequency, signal-to- noise-ratio). The simulation results show that the performance of this low-dimensional RW-KF of Per-path solution and Joint Multi Path RW-KF solution to the OFDM symbol which is used to estimate the channel condition. And also the simulation results show that the MSE performance of this low-dimensional Per-Path solution is comparable to that of the joint multi-path KF.

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