Abstract—Soft Iterative Decision Directed Channel Estimation (DDCE) employing Fast Data Projection Method (FDPM) subspace tracking algorithm and Variable Step Size Normalized Least Mean Square-based predictor is proposed for OFDM system in this paper. A more realistic Fractionally Spaced Channel Impulse Response (FS-CIR) model is considered. The performance of the soft iterative DDCE employing FDPM subspace tracking algorithm is validated in comparison with DDCE employing deflated version of Projection Approximation Subspace Tracking (PASTd) algorithm through computer simulations. The simulation results show that the soft iterative DDCE scheme based on FDPM algorithm outperformed its counterpart based on PASTd algorithm under both slow and fast fading channel scenarios. The proposed VSSNLMS-based CIR predictor for the iterative DDCE scheme also brings about an improved performance in comparison with the iterative scheme employing NLMS-based predictor.

Index Terms—Adaptive predictor, channel estimation, fading channel, iterative methods, OFDM system.

I. INTRODUCTION N the recent years much research efforts have been directed towards obtaining optimum and efficient channel estimation

techniques for deployment in single antenna-based OFDM system. This is because the availability of an accurate channel estimate is one of the major prerequisites for coherent symbol detection in an OFDM communication receiver. One of the robust techniques recently developed is the Decision Directed Channel Estimation (DDCE) scheme. Various works have been carried out regarding the DDCE scheme are well detailed in [1]. The DDCE scheme happened to have an edge over its competing counterpart, the pilot-assisted channel estimation method. This is because in the absence of symbol errors, the DDCE scheme can benefit from the availability of hundred percent pilot symbols by employing the detected symbols in combination with the sparsely available pilot symbols. However, in [2] the deflated version of Projection Approximation Subspace Tracking (PASTd) algorithm was proposed for the estimation of the Fractionally Spaced Channel Impulse Response (FS-CIR) for the DDCE scheme. The consideration for the channel model was hinged on the fact that FS-CIR model fits into a realistic channel condition

than Sample Spaced (SS)-CIR assumed in [3] since we have no control over the delay of the CIR taps. The FS-CIR model is also constituted by a relatively low number of statistically-independent Rayleigh fading paths in comparison with the SS-CIR [1]. As a result, the complexity of the DDCE scheme based on a realistic FS-CIR model reduces significantly compared with the one based on SS-CIR model.

In this paper we propose a soft input iterative DDCE scheme for OFDM systems. The scheme is based on a more robust subspace tracking algorithm, the Fast Data Projection Method (FDPM) subspace algorithm, for the implementation of CIR estimator module of the iterative DDCE scheme. Furthermore, we propose a Variable Step Size Normalized Least Mean Square (VSSNLMS)-based adaptive predictor for the implementation of the CIR predictor module of the scheme. The proposed iterative DDCE scheme is incorporated into the iterative loop of a Turbo decoder of the OFDM receiver, and it follows after the iterative techniques in [4-8].

In the next section, the system model and the fading channel model are described. In Section III, the iterative DDCE is presented with expressions for the proposed FDPM-based CIR estimator and VSSNLMS-based CIR predictor derived. Section IV contains the comparative performance results between the FDPM-based iterative DDCE and its counterpart, PASTd-based iterative DDCE, and the comparative results between the iterative DDCE scheme employing the proposed VSSNLMS-based CIR predictor and the scheme employing NLMS-based CIR predictor. Conclusion is finally given in Section V.

II. SYSTEM MODEL

A. OFDM Transceiver The block diagram of the OFDM transceiver is shown in

Fig. 1. At the transmitter, a binary signal is encoded by a Turbo encoder and interleaved by a random bit interleaver. The bits c′ are mapped to Quadrature phase shift keying (QPSK) constellation. The modulated signal y[n] is multiplexed with the QPSK modulated pilot symbols t[n]. The multiplexed symbols are serial-to parallel (S/P) converted and modulated by IFFT to form OFDM symbols x[n,k]. The OFDM symbols are transmitted over the frequency selective fading channel after the cyclic prefix (CP) has been appended. At the receiver the serial symbols are converted to parallel and

Soft Iterative Decision Directed Channel Estimation for OFDM Systems employing Adaptive Predictor

Olutayo Oyeyemi Oyerinde, Stanley Henry Mneney School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Durban,

4041, South Africa. Email: Mneneys@ukzn.ac.za and oyerinde@ukzn.ac.za

I

Acknowledgements: The financial support of the Deutscher AkademischerAustauschdienst/German Academic Exchange Service (DAAD) in the form ofDAAD/ANSTI postgraduate scholarship is gratefully acknowledged.

978-1-4244-4067-2/09/$25.00 © 2009 IEEE Wireless VITAE’09857

the cyclic prefix symbols are removed before taking the FFT of the remaining symbols. The iterative DDCE is employed to make an estimate of the channel state information (CSI). The estimate is fed to the soft demapper that uses the CSI in combination with the received message symbol to computes the soft information about each of the transmitted bits. As such, the Iterative DDCE and the soft demapper exchange information at every OFDM symbol time index n. Both iterative DDCE and soft demapper are, in turn, working in an iterative mode with the Turbo decoder during which they exchange soft information with Turbo decoder in a bid to refine the various outputs over a number of iterations. During the last iteration, the hard decided estimate of the transmitted bits ˆ

pb is made by the decoder.

B. Fading Multipath Channel Model In mobile wireless broadband communication, the

continuous-time CIR can be modeled as [1, 9] ( , ) ( ) ( )m m

mh t t cτ γ τ τ= −� , (1)

where ( )m tγ and mτ are the time-variant complex amplitude and the delay of the mth path respectively, and c(�) is the aggregate impulse response of the transmitter-receiver pair that corresponds to the square-root raised-cosine Nyquist filter. The channel dispersion could be characterized by maximum delay maxd mT τ� The Typical Urban delay profile in [10] is used in this work. However, due to the motion of one of the communicating terminals, ( )m tγ ’s are always modeled to be a WSS narrowband complex Gaussian processes which are independent for different paths. The frequency response of (1) at time t is described in [9] as

22( , ) ( , ) ( ) ( ) ,mj fj fm

mH t f h t e d C f t e π τπ ττ τ γ∞ −−

−∞= ��� (2)

where 2( ) ( ) j fC f c e dπ ττ τ∞ −−∞�� , is the Fourier transform pair of

the transceiver’s impulse response ( )c τ . However, for OFDM

systems with proper cyclic extension and timing the discrete subcarrier-related Channel Transfer Function (CTF) can be expressed as [9]:

0 1/

0 1[ , ] ( , ) [ , ] ( ) [ ] ,m s

K M k TklK m K

l mH n k H nT k f h n l W C k f nW τγ

−

= =Δ = = Δ� �� (3)

where 1

[ , ] ( , ) [ ] ( )M

s m s mm

h n l h nT lT n c lTγ τ=

= −�� , (4)

is the SS-CIR and exp( 2 / )π= −KW j K .The quantities M, K, K0 and Ts denotes the number of Fractionally Spaced (FS) channel paths, the number of OFDM subcarriers, the number of equivalent sample-spaced CIR taps, and the base-band signal’s sample duration respectively. In [1] it is noted that in realistic channel conditions associated with non-sample-spaced time-variant path-delays ( )m nτ , the receiver will encounter received signal components dispersed over several neighboring samples owing to the convolution of the transmitted signal with the system’s CIR. This is referred to as leakage. Generally, this phenomenon is unavoidable and therefore the resultant SS-CIR [ , ]h n l will comprise of numerous correlated non-zero taps as indicated in (1). In contrast to this, the FS-CIR ( ) ( )m mn nTγ γ= will be constituted by a low number of 0M K K≤ ≤ statistically independent non-zero taps associated with distinctive propagation paths [1]. As such, the complexity of the DDCE scheme based on a realistic FS-CIR model reduces significantly compared with the one based on SS-CIR model.

The received signal in the discrete frequency domain, after CP has been removed, is given as [1]

[ , ] [ , ] [ , ] [ , ]z n k H n k x n k w n k= + , (5) for 0,1,..., 1k K= − and all n’s, where [ , ]z n k ,

[ , ]x n k , [ , ]w n k and [ , ]H n k are the received symbol, transmitted symbol, additive white Gaussian noise sample and the complex CTF coefficient respectively, associated with the kth subcarrier of the nth OFDM block.

III. ITERATIVE DECISION DIRECTED CHANNEL ESTIMATOR The iterative DDCE consists of three special modules as shown in Fig. 2. The first part is the temporary CTF estimator which is referred to as a posteriori CTF estimator in [1, 2]. This makes a temporary estimate of the CTF coefficients

ˆ [ ]nH that correspond to the current channel state. Following this is the parametric CIR estimator that employs subspace tracking algorithm to generate the estimate of the current fractionally-spaced CIR. The last part is the CIR predictor or a prior CIR that produces an a priori estimate of the next CIR on a CIR tap-by-tap basis [1-3]. The predicted CIR is finally converted to the frequency domain CTF using the transformation matrixW� . The CTF obtained is then fed to the soft demapper for the purpose of obtaining soft information for the next OFDM symbol. The whole process then continued as described in Section II.

pb c [ ]ny

[n]t [ , ]n kx

z[n,k]

z[n,k]

H[n,k]�

y[n,k]

( )DaL c

( )SDeL c'

( )DeL c ( )SD

aL c'

c'

ˆ[ , ]y n k

( )SDaL c'

Fig.1. OFDM Transceiver with Iterative Decision Directed ChannelEstimator

858

A. MMSE-based Temporary CTF Estimator The Minimum Mean Square Error Estimator of [11] is used

to implement the temporary CTF estimator of the iterative DDCE scheme. This is similarly employed in [2]. The parameters H[n,k] is assumed to be complex-Gaussian

distributed with a zero mean and variance of 2Ησ . The noisy

MMSE estimate of the FD-CTF coefficients H[n,k] of the scalar linear model described by (5), using the two inputs information to the CTF Estimator, is then given as [1, 11]

1

2[n , k ] [n , k ] 1 [n , k ] [n , k ][n , k ] .2 2

w H w

y y y zH =σ σ σ

−� �� �+� ��

�

2

22

[n, k] [n, k]

[n, k] w

H

y z

y σσ

=+

, (6)

where the noisy estimate [n, k]H� could be written as [n , k ] [n , k ] [n , k ]H = H + v� , (7)

and [n,k]v denotes the i.i.d. complex-Gaussian noise samples

having a zero mean and a variance of 2v HMSEσ = . HMSE is

the Mean Square Error (MSE) associated with the MMSE CTF estimator of (6).

B. FDPM-based FS-CIR Estimator By substituting (3) into (7) we have [2]

/

1[ , ] ( ) [ ] [ , ]m s

M k Tm K

mH n k C k f n W v n kτγ

== Δ +�� , (8)

with ( )C f denoting the frequency response of the

transceiver’s pulse-shaping filter 12jKkW e

π−= , while mγ and

mτ are the amplitudes and the relative delays of the FS-CIR taps respectively. If (8) is written in matrix form we have

[ ] [ ] [ ]n n n= +� �H W vγ , (9)

where ( [ ])= diag C k�W W is defined as (K × M)-dimensional matrix in which ( [ ])diag C k is a (K × K)-dimensional diagonal matrix with the corresponding elements of the vector ( )C f on the main diagonal[1]. Symbol W is the Fourier Transform

matrix defined by m

s

kT

km KW Wτ

= for all K’s and M’s. Since the channel’s Power Delay Profile (PDP) and the associated transformation matrix ( )�W will be time-varying and might not be known a priori, if one of the communication terminals is in motion, the authors in [2] then proposed the PASTd subspace algorithm for tracking of this channel’s parameter.

Unfortunately, the PASTd algorithm possesses some limiting shortcomings. One of these is the fact that the deflated techniques that produces the PASTd algorithm results in a stronger loss of orthonormality between eigenvector [ ]m nw of

the transformation matrix n�W for ( )1,...,m M= [12-14].

Hence, another orthonormalization technique will be needed to be invoked for re-orthogonalization of the signal subspace after each update in order to extract the desired signal information. This process will of course result in increase of computational complexity. Another shortcoming is the fact that PASTd algorithm exhibits an increase in computational complexity if K M� [12]. In a bid to avoid these shortcomings, FDPM algorithm that possesses simple structure with a single parameter (the step-size) to be specified is hereby employed for accurate tracking of the FS-CIR’s PDP instead of PASTd algorithm. FDPM algorithm also has an extremely high convergence rate towards orthonormality, and is the fastest among all competing subspace algorithms of the same computational complexity [7]. The summary of the FDPM algorithm as applied to the tracking of the FS-CIR’s PDP is given in Table 1. Its performance index is defined in terms of the MSE criterion as follows:

{ }2( )MSE E n= e , (10)

where ( )ne is given as

ˆ ˆ( ) [ ] [ 1] [ ]n n n n= − −�e H W γ (11)

C. VSSNLMS -based CIR Predictor The CIR tap prediction approach derived in [15] is

employed in [2] to predict a priori estimate [ 1]m nγ +�of the

next CIR on a tap-by-tap basis. However in [16] the coefficient update complexity is said to be more costly for RLS-based predictor compared with NLMS-based predictor

TABLE I FAST DATA PROJECTION METHOD ALGORITHM

• Initialize [0]�W to orthonormal matrix (typically the first M columns of the identity matrix) with K rows.

• With the new observation ˆ [ ] [ ]n n= �H H , apply: For n = 1, 2, . . .

ˆˆ[ ] [ 1] [ ]n n n= −� HW Hγ (12a)

2ˆ [ ]nμμ =

H

(12b)

ˆ ˆ[ ] [ 1] [ ] [ ]Hn n n nμ= − ±�T W H γ (12c) 1ˆ ˆ[ ] [ ] [ ]n n n= −a eγ γ , (12d)

where [ ]1 10...0 T=e

( )22[ ] [ ] [ ] [ ] [ ]

[ ]Hn n n n n

n= −Z T T a a

a (12e)

{ }[ ] [ ]n normalize n=�W Z , (12f) where normalize{.} is the normalization of each column of [ ]nZ for all M.

end

[ , ]n kz[ , ]n ky

ˆ [ ]nH ˆ[ ]nγ [ 1]nγ +� [ 1]n +�

H

[ ]n� HW

ˆ[ , ]n ky

Fig. 2. Iterative Decision Directed Channel Estimator [2]

859

especially for large filter lengths. It is against this background that we seek to derive an improved version of NLMS-based predictor with faster convergence rate close to, but with less complexity compared with RLS-based predictor for the proposed iterative DDCE scheme. The derivation of the Variable Step Size NLMS (VSSNLMS) predictor follows after the NLMS-based CIR predictor of [15].

Using the NLMS algorithm, the CIR tap’s predictor filter coefficients [ ]m np of length Lprd are updated as [15]

*2

ˆ[ ] [ 1] [ ] [ 1]ˆ [ 1]

m m m mm

n n e n nn

μ= − + −−

p p γγ

, (13)

where μ is the constant step size. For stable operation the NLMS algorithm requires 0 2μ< < [16]. Using the method used to derive the VSSNLMS-based channel estimator for Turbo Equalizer-based communications receiver in [8], the VSSNLMS-based predictor is hereby derived as follows:

*2

[ ] ˆ[ ] [ 1] [ ] [ 1]ˆ [ 1]

m m m mm

nn n e n nn

μ= − + −−

p p γγ

(14)

ˆ[ 1] [ ] [ ]Hm m mn n nγ + =� p γ , (15)

where ˆ ˆ ˆ[ ] [ ] [ ] [ ] [ 1] [ 1]H

m m m m m me n n n n n nγ γ γ= − = − − −� p γ (16) is the prediction error and [ ]nμ is the variable step-size which is updated following [17] as

2[ ]ˆ[ ] [ 1]2 [ 1]

e nn nn

ρμ μμ∂= − −

∂ − (17a)

2 [ ][ ]ˆ[ ] [ 1] .2 [ ] [ 1]

m

m

ne nn nn n

ρμ μμ∂∂= − −

∂ ∂ −p

p (17b)

*

2

ˆ ˆ[ , ] [( 1), ] [ ] [ 1]ˆ[ ] [ 1]ˆ [ 1]

Tm m

m

e n k e n k n nn n

n

ρμ μ

− −= − +

−

γ γ

γ. (18)

For the complex input signal and complex channel tap, (18) becomes [18]:

{ }*

2

ˆ ˆ[ , ] [( 1), ] [ ] [ 1]ˆ[ ] [ 1] .

ˆ [ 1]

Hm m

m

e n k e n k n nn n

n

ρμ μ

ℜ − −= − +

−

γ γ

γ(19)

In order to restrict the variable step size [ ]nμ to the range 0 [ ] 2nμ< < which makes for stable operation of NLMS algorithm as stated above, the variable step size [ ]nμ in (14) is then confined to within the range given as

max max

min min

ˆˆ[ ]

ˆ[ ]

ifn if

n otherwise

μ μ μμ μ μ μ

μ

>�= <��

, (20)

where min max0 2μ μ< < < . For n = 0, the prediction filters are initialized as [15] [ ][ ] 1 0 0 . . . 0 T

m n =p .

IV. SIMULATION RESULTS Simulations have been run to confirm the performance of

the proposed iterative DDCE scheme employing FDPM

subspace tracking algorithm and VSSNLMS predictor. A rate 1/3, octal generator polynomial of (7, 5), turbo-coded QPSK-modulated OFDM system with K = 64 subcarriers and a total bandwidth of 800kHz is assumed. The symbol duration, Ts is 80μs, while the CP length is 16 samples (1/4 of the symbol period) with the CP period, Tg = 20μs. As a result the total block period, T is 100μs. The six-path time-varying Rayleigh fading COST 207 Typical Urban (TU) channel model of [10] with Doppler frequencies of 50Hz and 100Hz is employed. The first OFDM symbol, of the 64 subcarriers, comprises of the pilot symbols. These are used for the initialization of the channel estimation process. In all the simulation, we assumed having M = 6 FS-CIR taps, and we set μ = 0.98 for FDPM algorithm, while η is set to 0.95 for PASTd algorithm. The length of the CIR predictor (Lprd) is set to 10, while μ0 = 0.5 is used for NLMS-based predictor and to initialize VSSNLMS-based predictor, ρ is set to 0.002.

The Soft Demapper presented in [19] is adapted for use in the iterative mode with the proposed iterative channel estimation. The two Soft Mappers follow after the one in [20]. The Soft Mapper-1 is similar to Soft Mapper-2 except that its a priori values are set to zero since there is no feedback from the decoder to Soft Mapper-1. Perfect time and frequency synchronization is assumed in all the simulations in this paper.

Fig. 3 shows the comparative performance gain of FDPM-based iterative DDCE over PASTd-based iterative DDCE in the form of BER for both slow and fast fading scenarios respectively. The curves labeled perfect channel state information (CSI) correspond to detection with perfect knowledge of channel at the receiver, and serve as benchmark in the two cases of channel fading. The figure also indicates the performance improvement introduced by the employment of the proposed VSSNLMS predictor over its NLMS counterpart. The corresponding Means Square Error (MSE) results are shown in Fig. 4 and Fig. 5 for slow and fast fading channel respectively. From the results, it seems there is no significant improvement in the performance of the DDCE employing VSSNLMS predictor in comparison with the one with NLMS predictor during the fast fading channel scenario. This might be due to the fact that improvement brought to the scheme by the iterative techniques override the improvement the VSSNLMS predictor offers. Hence, the performances of the schemes employing the two predictors closed up during the fast fading scenario.

V. CONCLUSION In this paper, we have proposed an iterative DDCE scheme

employing FDPM subspace algorithm. A VSSNLMS-based predictor is also proposed for the implementation of the adaptive predictor module of the iterative DDCE scheme. The simulation results show a significant improvement in the performance of the proposed iterative scheme employing the FDPM algorithm over its counterpart that employs the PASTd algorithm. The results also indicate that the VSSNLMS-based CIR predictor brings about a significant improvement to the performance of the DDCE scheme in comparison with the

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DDCE scheme employing NLMS-based predictor, especially during slow channel fading scenario.

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0 1 2 3 4 5 6 7 8 9 1010

-5

10-4

10-3

10-2

10-1

100

SNR[dB]

BE

R

FDPM-w ith-VSSNLMS: fD = 0.005

FDPM-w ith-NLMS: fD = 0.005

PASTd-w ith-VSSNLMS: fD = 0.005

PASTd-w ith-NLMS: fD = 0.005

FDPM-w ith-VSSNLMS: fD = 0.02

FDPM-w ith-NLMS: fD = 0.02

PASTd-w ith-VSSNLMS: fD = 0.02

PASTd-w ith-NLMS: fD = 0.02

Perfect CSI

Fig.3. BER at the 7th iteration for the proposed DDCE-based FDPM and PASTdalgorithms, employing NLMS and VSSNLMS predictors, Fd = 0.005 and Fd =0.02

0 1 2 3 4 5 6 7 8 9 10-20

-18

-16

-14

-12

-10

-8

-6

-4

SNR[dB]

MS

E[d

B]

7th Iter.,PASTd-with-VSSNLMS-based Iter.DDCE

7th Iter.,FDPM-with-VSSNLMS-based Iter.DDCE

7th Iter.,PASTd-with-NLMS-based Iter.DDCE

7th Iter.,FDPM-with-NLMS-based Iter.DDCE

Fig.4. MSE at the 7th iteration for the proposed DDCE-based FDPM andPASTd algorithms, employing NLMS and VSSNLMS predictors, Fd = 0.005

0 1 2 3 4 5 6 7 8 9 10-16

-14

-12

-10

-8

-6

-4

-2

SNR[dB]

MS

E[d

B]

7th Iter.,PASTd-with-VSSNLMS-based Iter.DDCE

7th Iter.,FDPM-with-VSSNLMS-based Iter.DDCE

7th Iter.,PASTd-with-NLMS-based Iter.DDCE

7th Iter.,FDPM-with-NLMS-based Iter.DDCE

Fig.5. MSE at the 7th iteration for the proposed DDCE-based FDPM andPASTd algorithms, employing NLMS and VSSNLMS predictors, Fd = 0.02

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