 Open Access
 Total Downloads : 6
 Authors : K. Vidhya, M. Geetha
 Paper ID : IJERTCONV3IS12057
 Volume & Issue : NCICCT – 2015 (Volume 3 – Issue 12)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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 timedomain correlation of the channel by employing a Kalman filter based on an approximation of the timevarying 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 RWKF 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 multipath Kalman gain formulation through the Woodburys identities. Hence, this new algorithm consists of a superposition of independent singlepath singlecarrier KFs. This observation allows us to adapt the optimization to the actual multipath 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, signaltonoiseratio). These analytic formulae are given for the first, second, and thirdorder RW models used in the KF. The proposed perpath KF is shown to be as efficient as the exact KF (i.e., the joint multipath KF).
KeywordsOrthogonal frequency division multiplexing; Channel estimation; Least square estimator; Random walk model; Kalman filter.

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 frequencydivision multiplexing (FDM) scheme used as a digital multicarrier 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 subcarrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation or phaseshift keying) at a low, symbol rate maintaining total data rates similar to the conventional Singlecarrier modulation schemes in the same bandwidth.
The primary advantage of OFDM over singlecarrier schemes is its ability to cope with severe channel conditions (for example, attenuation of high frequencies in a long copper wire, narrowband interference and frequencyselective 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 signaltonoise 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 singlecarrier system.
Orthogonal FrequencyDivision Multiplexing (OFDM) is an effective technique for alleviating frequencyselective channel effects in wireless communication systems. In this technique, a wideband frequencyselective channel is converted to a number of parallel narrowband flat fading sub channels which are free of InterSymbolInterference (ISI) and free of InterCarrier 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.

PROBLEM FORMULATION
Most of the conventional methods work in a symbolby 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 timedomain correlation. This paper deals with channel multipath Complex Amplitude (CA) estimators based on KFs. An approximation often used in the literature consists of approaching the fading process as autoregressive [4]. Hence,
a widely used channel approximation is based on a firstorder AutoRegressive 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 multipleinputmultiple 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 quasioptimal 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 highspeed channel). However, here we consider moderate normalized Doppler frequency (fdT)
values; i.e., fdT102. 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 102 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 meansquare error (MSE) performance for a given Doppler and signaltonoise ratio (SNR) scenario. This performance can be obtained by a firstorder Random Walk (RW)model based KF (RW1KF) ([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 multipath values where L is the number of multipaths and applies a KF to each path.

SYSTEM DESCRIPTION

OFDM System Model
Orthogonal frequency division multiplexing (OFDM) is a parallel transmission scheme, where a highrate serial data stream is split up into a set of lowrate sub streams, each of which is modulated on a separate subcarrier (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 kth OFDM symbol. X(k) transmitted symbol is modulated by (MPSK). Modulate symbol transmitted on the subcarrier with indice n1N/2. The sequence of transmitted symbol is assumed to be zero mean and stationary with normalized variance. After transmission over a slowly timevarying multipath channel and fast Fourier transform demodulation, the kth 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 lth path at kth OFDM symbol.

Pilot Pattern
We use Np pilot subcarriers, they are evenly inserted into the N subcarriers. The received pilot subcarriers 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 NpÃ—L matrix Fp is the Fourier
the integrated Brownian model) for the approximation model.
A secondorder RW model and a thirdorder 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 RW2KF and RW3KF estimators, respectively. The RWKF estimators of the previously cited studies were designed for singlepath
[Fp]np,l = e (4) 
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 singlecarrier systems. In the present study, we consider multipath 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
highdimensional 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

Single Path RWKF
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 lth 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 perpath KF, denoted by w en, the statespace del of per h KF for the lth s given
ath
ath
^ (
ath
ath
^ (

Joint MultiP
mates
/ )
mates
/ )
=
=
Joint multip CA to esti
RWKF
RW
/ )
RW
/ )
by
by
u
)
u
)
Wood
( )
Wood
( )
b
b
ositio ury
( )
ositio ury
( )
KF super p ning the perpath 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 tesp model of the perpath KF for the lth path CA ca applied to singlepath RWKF which
( (
/
)
S
/
)
S
=
(SP
=
(SP
( (
( (
P^ M P^
( )=
( )=
P
P
S
S
( )
( )
/
(11)
+
M
+
M
( )
( )
( )
( )
(
(
U 12)
/
/
)
)
estimates ^ / ) he state model of the Lpath 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 lth path, u(k) = [u( w . u( ] with u(
the state noise vector of the lt th, ILÃ—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 Â½] [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 Â½] [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 = ILÃ—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. Perpath KF structure.
This CA is compared with the CA of OFDM transmitted symbol X(k). If the CA of joint multi path RWKF 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 RWKF 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 RWKF is equal to the CA of OFDM transmitted symbol
the joint multipath 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 lognormal
K( P(
S /S P(
S + (12) distribution with a standard deviation according to the log
( ( (
distance path loss model.

Asymptotic Mean Square Error of the PerPath 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 RWKF allows us to provide analytic formulas for the mean square error performance. The MSE performance can still be improved by proposed estimator.


SIMULATIONS
In this simulation section, The PSKOFDM system with N=128 subcarriers 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 RWKF 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 RWKF, MSE of Per Path KF Versus SNR, MSE comparison between Per Path and Joint Multipath RWKF 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 RWKF signal.
Fig. 2 shows the input (Xk) signal with 128 subcarriers 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 subcarriers 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 subcarriers. Fig. 6 presents the OFDM symbol with pilot subcarriers which is generated by the Fourier matrix and diagonal elements of input signal with pilot subcarriers.
Fig. 9. Joint Multi path RWKF signal.
Fig. 10. MSE Of Joint Multi Path RWKF And Per Path RWKF.
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 singlepath RWKF and Joint multipath RWKF respectively. This joint multipath CA is comparable with the input signal CA to analyze the Multipath channel condition. Fig. 10 presents MSE of Joint Multi Path RWKF and Per Path RWKF. 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 RWKF
Fig. 12 shows the MSE versus fdT and Fig. 13 shows MSE comparison between Per Path and Joint Multi Path RWKF.

CONCLUSION
This paper proposed the integrated RWKF for channel condition estimation. Our solution is a twostep 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 timedomain correlation of the channel, while the LS step exploits the frequencydomain correlation of the channel. We have shown how to apply the previous results we obtained for a singlepath singlecarrier to the multipath multicarrier context. Lower dimensional RWKF 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 multipath KF. This has been done through the Woodburys identities. Hence, this new algorithm consists of a superposition of independent singlepath single carrier KFs, which were allows us to adapt the optimization to the actual multipath multicarrier 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, signalto noiseratio). The simulation results show that the performance of this lowdimensional RWKF of Perpath solution and Joint Multi Path RWKF 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 lowdimensional PerPath solution is comparable to that of the joint multipath KF.
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