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Doppler Spread Estimation in Frequency Selective Rayleigh Channels

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Doppler Spread Estimation in Frequency Selective Rayleigh Channels

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									     Doppler Spread Estimation in Frequency
 Selective Rayleigh Channels for OFDM Systems
                                         Athanasios Doukas, Grigorios Kalivas
                                           {adoukas, kalivas}@ee.upatras.gr
                          Department of Electrical and Computer Engineer, University of Patras
                                        Campus of Rion, Achaia, 26500, Greece


   Abstract- In this paper, we present a method for              In addition, in OFDM systems, if the channel varies
estimating the Doppler spread (DS) in Wireless Local Area        considerably within one OFDM symbol because of high
Networks (WLAN) using Orthogonal Frequency Division
Multiplexing (OFDM). DS gives a measure of the fading rate       MU mobility, orthogonality between subcarriers is lost,
of the wireless channel, which can be used to adjust the         leading to inter-carrier interference (ICI) [4], degrading
channel estimation rate and create specifically designed         system performance. Doppler information can help in
channels estimators to combat Inter-Carrier Interference         selection of appropriate transmission characteristics to
(ICI) induced due to loss of orthogonality that DS imposes on    combat ICI including proper channel estimators to
OFDM systems. The estimation is based on the
autocorrelation function of time domain channel estimates        enhance reception. Specifically designed channel
over two OFDM symbols and since that most of the receiver        estimators can be applied and the rate of appliance can be
algorithms require knowledge if the receiver moves or not we     chosen to improve throughput. When mobility is low the
divide the operation region into two modes: still mode(S-        rate of estimation can be lowered and increase throughput
mode) and moving mode (M-mode). The estimation accuracy,         this way. In the other way, an increase in the estimation
examined in environments with different PDPs, including
channel sparsity, using several constellation schemes is quite   rate can help to lower BER.
accurate from low SNR values of 5 dB.                               Doppler spread estimation is widely examined in the
                                                                 literature. Level Crossing Rate (LCR) of the received
Keywords: Doppler estimation, fading channels.
                                                                 signal envelope is proportional to the Doppler frequency
                                                                 [5] and thus used in Doppler estimation. However, the
                    I.   INTRODUCTION                            fading nature of the wireless channel decreases the
   Orthogonal Frequency Division Multiplexing (OFDM)             estimator’s accuracy in low Doppler values. Other
[1] has been widely applied in the last years for various        methods, highly associated to LCR, are the Zero Crossing
wireless communication systems such as Digital Video             Rate (ZCR), which uses the in-phase or quadrature phase
Broadcasting (DVB) and wireless local area networks              (I/Q) signal part [6], and some other higher order
(WLANs) ensuing great success. These systems however,            crossings of the signal envelope [7]. Switching rate of
should be capable of working efficiently in wide range of        diversity branches is used for the velocity estimation in [8],
operating conditions, such as large range of mobile unit         but in [9] its sensitivity to the fading channel is shown. In
(MU) speeds, different carrier frequencies in licensed and       [10] and [11], velocity estimation algorithms that use
un-licensed bands, various delay spreads, and wide               pattern recognition and wavelets respectively are proposed.
dynamic signal to-noise ratio (SNR) ranges. This way             Another type of estimators are the covariance-based
assessing the channel quality and its rate of change is of       estimators, which estimate the Doppler frequency from
great importance in adapting the system parameters to            the autocovariances of powers of the received signal
continuously changing channel conditions [2], [3].               envelope [12], or from sums of the I/Q components [13].
   The previously mentioned reasons motivated the use of         The estimators presented in [14], based on the second
adaptive algorithms in wireless communication systems.           spectral moment of the I/Q components, and the
Scope of the adaptive algorithms is to optimize wireless         correlation-based estimators in [15] and [16], share the
systems performance, fully exploit channel capacity and          same basis.
utilize available resources into the most efficient manner          In all the previous works, crossing-rate estimators are
to maximize their throughput for a given quality of service      less efficient than their covariance-based counterparts
(QoS), which in most of times is measured in terms of Bit        over short estimation windows, due to the fact that the
Error Rate (BER). However, adaptation necessitates the           observed signal does not experience many level crossings
accurate knowledge of some wireless channel’s                    in a small time window [14]. This observation is
parameters.                                                      important for practical applications, as a wireless channel
    One crucial parameter in adaptation of wireless              can only be assumed to be wide-sense stationary (WSS)
systems is the maximum Doppler spread. It provides               over short intervals.Furthermore most of the estimators
information about the fading rate of the channel.                suffer from insufficient performance in low Doppler
Knowledge of Doppler spread can improve detection and            values and for low SNR values and are computationally
aid into transmission optimization in both physical layer        high, needing a significant load of measurements.
and higher levels of the protocol stack [2].In particular, a        In the previous works OFDM is used in DVB systems,
power control update approach can be applied, adjust of          where the Doppler frequency is high. Yet in WLAN
interleaving length to reduce reception delays and so on.        systems the mobility is lower, ranging from 0 to 40Hz.
Thus the most critical for a WLAN system is to know                                  ∞

whether is standing still (S-mode) or moving (M-mode)                  H ( t , ƒ ) ≡ ∫ h ( t , τ)e − j2 π ƒ τ dτ = ∑ γ l ( t )e − j2 π ƒ τl . (3)
and when moves if retains its velocity. Such an estimator                            −∞                              l

is presented only in [17], where the operation mode is
divided into three different states, of low, medium and fast          Hence, the correlation function of the frequency
mobility.                                                          response for different times and frequencies is
   In this work we present a Doppler estimator that uses
the time correlation of only two OFDM symbols,                                                         {
                                                                              rH (∆t, ∆ ƒ ) ≡ Ε Η (t + ∆t , ƒ + ∆ ƒ )Η * ( t ,ƒ )        }
satisfying the WSS channel assumption and achieving low
                                                                                               = ∑ rγ l (∆t )e    − j2 π∆ ƒ τl
complexity, which manages to clearly distinguish the two                                           l
modes of mobility. A Doppler estimator that achieves                                                                                                (4)
                                                                                                         ⎛                      ⎞
these two goals, to the best of authors’ knowledge, is the                                     = rt (∆t )⎜ ∑ σ l e − j2 π∆ ƒ τl ⎟
                                                                                                                2

first time that is presented. We examine its estimation                                                  ⎝ l                    ⎠
performance using various constellation schemes with                                           = σΗ rt (∆t )rƒ (∆ ƒ )
                                                                                                    2
different bit rates in environments with a variety of Power
Delay Profiles (PDP), including channel sparsity. The
                                                                   where σ H = 1 is the total average power of the channel
                                                                           2
estimator in most of the cases works satisfactory from
SNR values as low as 5 dB.                                         impulse response defined as
   The rest of the paper is organized as follows. In Section
II the channel model along with the OFDM system used                                             σΗ ≡ ∑ σl
                                                                                                  2      2
                                                                                                                                             (5)
are described. The estimator and its characteristics are                                                   l

derived in Section III. Estimation results are presented in                                                σ l − j2 π∆ ƒ τl
                                                                                                             2

Section IV. Finally concluding remarks are given in
                                                                                      rƒ (∆ ƒ ) = ∑            e                             (6)
                                                                                                       l   σΗ
                                                                                                            2

Section V.
                                                                                                                        2
       II. FADING CHANNEL AND OFDM SYSTEM                             Without loss of generality, we also assume that σ Η = 1 ,
                                                                   which, therefore, can be omitted from (4).
   Prior to examining Doppler estimation for OFDM                     From (4), the correlation function of H(t,ƒ) can be
systems in WLAN radio channels, we briefly describe the            separated into the multiplication of a time domain
channel statistics, emphasizing the separation property of         correlation rt(∆t) and a frequency domain correlation
wireless channels, which is crucial employing our                  rƒ(∆ƒ). rt(∆t) is dependent on the vehicle speed or,
estimator. In this section we also describe the OFDM               equivalently, the Doppler frequency, while rƒ(∆ƒ) depends
system used for our estimator.                                     on the multipath delay spread. With the separation
A. Wireless Channel Model                                          property, we are able to propose our Doppler estimator
  The complex baseband representation [18] of a wireless           described in the next section.
channel impulse response can be described by                          For an OFDM system with block length Tƒ and tone
                                                                   spacing (subchannel spacing) ∆ƒ, the correlation function
                                                                   for different blocks and tones can be written as
              h ( t , τ) ≡ ∑ γ l ( t )δ(τ − τl )             (1)
                            l
                                                                                            rΗ [i, k ] = rt [i]rƒ [k ]                       (7)
where τℓ is the delay of the ℓ-th path and γℓ(t) is the
corresponding complex amplitude and δ is the delta                 where
function. Due to the motion of the vehicle, γℓ(t)’s are
wide-sense stationary (WSS) narrowband complex                                                 rt [i] ≡ rt (iTƒ )                            (8)
Gaussian processes, which are independent for different                                       rƒ [k ] ≡ rƒ (k∆ ƒ )                           (9)
paths.
  We assume that γℓ(t) has the same normalized
correlation function rτ(∆t) for all ℓ and, therefore, the          and rt and rƒ are the time and frequency correlation
                                                                   respectively. Equation (7) is valid for an exponentially
same normalized power spectrum pt(Ω). Hence
                                                                   decaying multipath PDP, which is used in the simulations.

                      {              l    }
        rγ l (∆t ) ≡ Ε γl (t + ∆t )γ * ( t ) = σl rt (∆t )
                                                2
                                                             (2)
                                                                   Exponential PDP is the most commonly accepted and
                                                                   accurate model for indoor channels [19].
                                                                      From Jakes’ model
where σl is the average power of the k-th path.
        2

  Using (1), the frequency response of the time-varying                                   rt [n ] = J 0 (nωd ) ≡ rJ [n ]                   (10)
radio channel at time t is
                                                                   where J0(x) is the zeroth-order Bessel function of the first
                                                                   kind, and its Fourier transform (FT) is
               ⎧2       1
               ⎪ω               ,                        if ω < ωd
     p J (ω) = ⎨ d 1 − (ω ωd )2                                      (11)
               ⎪
               ⎩0, otherwise

   In the above expression ωd=2πΤƒƒd, and ƒd is the
Doppler frequency, which is related to the vehicle speed
υ and the carrier frequency ƒc by

                                       υƒ c
                              ƒd =                                   (12)
                                        c

where c is the speed of light.
B. OFDM System                                                                                 Figure 1. OFDM Baseband Model

   The baseband model of the OFDM system employed
                                                                                                                       N r −1− ρ
into this paper is shown in Fig. 1. Time domain samples of                                                1
an OFDM symbol can be obtained from data symbols as
                                                                                           rt [ρ] =
                                                                                                        Nr − ρ
                                                                                                                         ∑Υ
                                                                                                                         i =0
                                                                                                                                   i ,k   ⋅ Υi* ρ ,k
                                                                                                                                              +
                                                                                                                                                                (16)


        x i ,n = IFFT{X i ,k }                                              where Nr is the number of used samples and ρ is the time
                  N −1                                               (12)   difference of the used samples.
               = ∑ X i ,k e j2 πnk / N       n = 0, K , N − 1                  Firstly we set Nr=2 and ρ=0 in (10) and (16). Thus we
                  k =0
                                                                            get

where Xi,k is data symbol of the k-th subcarrier of the i-th                                rt [0] = J 0 (0 ⋅ ω d ) = J 0 (0 ) = 1                              (17)
OFDM symbol, and N is the number of subcarriers. After
                                                                                                 2−1− 0                                                   1
the guard interval addition, the samples xi,n are transmitted                              1                                                           1
over the linear time dependent channel described in eq. 1 ,
                                                                               rt [0] =
                                                                                          2− 0
                                                                                                   ∑Υ
                                                                                                   n =0
                                                                                                             i ,k   ⋅ Υi* 0 ,k ⇒ rt [0] =
                                                                                                                        +                                ∑ Υi,k ⋅ Υi*,k
                                                                                                                                                       2 i =0     (18)
with additive white Gaussian noise (AWGN) zi,n, with
zero mean and variance of σ 2 . In this paper, we assume
                               z                                              It is clear that this step is actually a normalization
the channel to be constant over an OFDM symbol, but                         purpose step. In the next step we set Nr=2 and ρ=1 in (10)
time-varying across OFDM symbols, which is a                                and (16). Thus we get
reasonable assumption for low mobility.
   At the receiver, assuming perfect synchronization, the                                  rt [1] = J 0 (1* ωd ) = J 0 (2πTƒ ƒ d )                              (19)
received samples can be expressed as
                                                                                               2−1− 1
                                                                                          1
                   y i , n = x i , n ∗ h ( t , τ) + z i ,n           (13)
                                                                              rt [1] =
                                                                                         2−1
                                                                                                 ∑Υ
                                                                                                 i =0
                                                                                                          i ,k   ⋅ Υi* 1 ,k ⇒ rt [1] = ∑ Υ0,k ⋅ Υ1*,k (20)
                                                                                                                     +




where * stands for convolution.                                               Combining (17) to (20) we get Doppler frequency by
  After removing the guard interval, the receiver de-
multiplexes the received samples by using the FFT as                                                   ⎛                                           ⎞
                                                                                                       ⎜                                           ⎟
                                                                                                 1 −1 ⎜ ∑ Υ0,k ⋅ Υ1,k
                                                                                                                    *
      Yi ,k = FFT{y i ,n }                                                                 ƒd =     J0                                             ⎟            (21)
                                                                                                2πTf ⎜ 1 1                                         ⎟
                                                                                                       ⎜ ∑ Υi ,k ⋅ Υi ,k
                                                                                                                      *
               1 N−1                                                 (14)                                                                          ⎟
           =     ∑ yi,n e− j2πnk / N
               N n =0
                                                k = 0, K, N − 1                                        ⎝ 2 i =0                                    ⎠

                                                                                               IV. SIMULATION RESULTS
and the demodulated signal Yi,k can be expressed as
                                                                               In this section we are going to describe the simulation
                 Yi ,k = X i ,k ⋅ H( t ,ƒ ) + Zi ,k .                (15)   results of the Doppler Estimator. We have tested the
                                                                            estimator in BRAN Channels A and B [20], which are
                                                                            typical for indoor WLANs. These channels are quite harsh
            III. DOPPLER SPREAD ESTIMATOR                                   and include sparsity.
                                                                               In Fig. 2 the estimation results for BRAN Channel A
   In this section we describe the functions used to derive
                                                                            with 5 dB SNR using BPSK modulated data for different
the Doppler estimator estimator. We are going to use the
                                                                            Doppler Spread values, ranging from 0 to 40 Hz, are
time correlation function expressed in (10) and the time
                                                                            depicted. From this figure we can see that the scope of the
correlation function using the received samples expressed
                                                                            estimator, to clearly distinguish the two operational modes,
by the following equation
                                                                            is succeeded. Especially from subcarriers 5 to 35, the
                                                                            distinction is clearer.
                                                                                                                                                              -3   Doppler estimation with SNR=10dB in BRAN Channel A for QPSK modulated data
                             -3
                                  Doppler estimation with SNR=5dB in BRAN Channel A for BPSK modulated data                                            x 10
                      x 10                                                                                                                   7.5
                  8



                  7
                                                                                                                                               7
                  6


                  5                                                                                                                          6.5




                                                                                                                         E s t im a t io n
     Estimation




                  4
                                                            Doppler estimation   0 Hz
                                                            Doppler estimation   10 Hz
                                                            Doppler estimation   20 Hz                                                         6                                                    Doppler estimation 0 Hz
                  3                                         Doppler estimation   30 Hz                                                                                                              Doppler estimation 10 Hz
                                                            Doppler estimation   40 Hz
                                                                                                                                                                                                    Doppler estimation 20 Hz
                  2                                                                                                                                                                                 Doppler estimation 30 Hz
                                                                                                                                             5.5                                                    Doppler estimation 40 Hz

                  1


                  0                                                                                                                            5
                      0            5       10       15      20         25         30     35    40       45    50                                   0                5       10       15       20         25         30         35   40     45   50
                                                                 Subcarrier Index                                                                                                                  Subcarrier Index


 Figure 2. Doppler estimation results in BRAN Channel A with BPSK                                                   Figure 4. Doppler estimation results in BRAN Channel A with QPSK
                                                modulation with 5dB SNR.                                                                                                         modulation with 5dB SNR
                                                                                                                                                          -3       Doppler estimation with SNR=10dB in BRAN Channel B for BPSK modulated data
                             -3
                                  Doppler estimation with SNR=5dB in BRAN Channel A for BPSK modulated data                                        x 10
                      x 10                                                                                                                     8
                  8


                                                                                                                                               7
                  7


                                                                                                                                               6
                  6
                                                                                                                                                                                                         Doppler estimation 0 Hz
                                                                                                                                               5                                                         Doppler estimation 10 Hz
                  5                                                                                                                                                                                      Doppler estimation 20 Hz
                                                                                                                             E s tim a tio n
                                                                                                                                                                                                         Doppler estimation 30 Hz
     Estimation




                                                                                                                                               4                                                         Doppler estimation 40 Hz
                  4
                                                            Doppler estimation   0 Hz
                                                            Doppler estimation   10 Hz
                                                            Doppler estimation   20 Hz                                                         3
                  3                                         Doppler estimation   30 Hz
                                                            Doppler estimation   40 Hz
                                                                                                                                               2
                  2

                                                                                                                                               1
                  1

                                                                                                                                               0
                  0                                                                                                                                0               5       10       15        20         25         30         35   40     45   50
                      0            5       10       15      20         25         30     35    40       45    50
                                                                 Subcarrier Index
                                                                                                                                                                                                   Subcarrier Index


                                                                                                                     Figure 5. Doppler estimation results in BRAN Channel B with BPSK
 Figure 3. Doppler estimation results in BRAN Channel A with BPSK
                                                                                                                                                                                            modulation .
                                             modulation with 10 dB SNR.


   Analogous results are taken in the case of 10 dB SNR,                                                           are presented. Likewise the case of QPSK data in Fig. 4,
depicted in Fig. 3. In this figure, the distinction between                                                        the estimator can not distinguish the two modes of
the two modes is clearer because of the higher SNR. From                                                           operation. This result is due to the harsher transmission
these two figures and taking under consideration that in                                                           conditions that are imposed into this case. BRAN Channel
most of the WLAN standards such as 802.11a,                                                                        B is a harsher transmission environment than BRAN
HIPERLAN/2, there is a part of two BPSK modulated                                                                  Channel A. Thus the noise added in the received signal
useful symbols in the preamble part, we can say that this                                                          results to insufficient estimation results. Again the
method can be applied on the preamble data and                                                                     insertion of a channel estimator would probably improve
distinguish the two mobility states. This way the system                                                           the estimation results.
will be able to adapt its transmission scheme before the
                                                                                                                                                                                   V. CONCLUSIONS
useful packet of data is transmitted and thus increase its
performance.                                                                                                          We have presented a Doppler estimator for low
   In Fig. 4 the estimation results for BRAN Channel A                                                             mobility OFDM systems in Frequency Selective Rayleigh
with 5 dB SNR using QPSK modulated data for different                                                              Fading Channels, using only two OFDM symbols for the
Doppler Spread values, ranging from 0 to 40 Hz, are                                                                time correlation in wireless OFDM systems. The estimator
depicted. In this case even that the estimator manages to                                                          instead of trying to estimate the accurate value of the
slightly distinguish the different Doppler spreads, the                                                            Doppler frequency divides the mobility into a still and a
difference is negligible. This result can be explained from                                                        moving mode, which for the case of WLANs is the most
the constellation type used. QPSK is more sensitive to                                                             important. We have examined its performance in wireless
noise than BPSK. Thus when the correlation of the                                                                  channels with different power delay profiles, including
received signal is estimated, the presence of noise has a                                                          sparse channels. The estimator, in most of the cases,
significant effect on its value leading to insufficient                                                            manages to clearly distinguish the two modes of mobility,
estimation. A channel estimator that would remove part of                                                          still or moving.
the noise would probably improve the estimation results.
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