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I. IN

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Inspired by re-weighted ℓ -norm minimization algorithm in [6], in this paper, we propose an improved ASCE method using re-weighted zero-attracting NLMF (RZA-NLMF) algorithm. Unlike the ZA-NLMF algorithm [5], RZA-NLMF algorithm can penalize different magnitudes of channel coefficients with different sparse constraint strength. The effectiveness of our proposed method is confirmed by computer simulation.

The remainder of this paper is organized as follows. A system model is described and standard LMF and NLMF algorithms are introduced in Section II. In section III, sparse ASCE using ZA-NLMF algorithm is introduced and improved ACSE using RZA-NLMF algorithm is highlighted. Computer simulations are presented in Section IV in order to evaluate and compare performances of the proposed ASCE methods. Finally, we conclude the paper in Section V.

II. SYSTEM MODEL AND STANDARD LMF ALGORITHM

Consider a baseband frequency-selective fading wireless communication system where FIR sparse channel vector = [ℎ , ℎ , … , ℎ ] is -length and it is supported only by

nonzero channel taps. Assume that an input training signal ( ) is input to probe the unknown sparse channel. At the receiver side, observed signal ( ) is given by ( ) = ( ) + ( ), (1)

where ( ) = [ ( ), ( − 1), … , ( − + 1)] denotes the vector of training signal ( ) , and ( ) is the additive white Gaussian noise (AWGN) assumed to be independent with ( ). The objective of ASCE is to adaptively estimate the unknown sparse channel estimator using the training signal ( ) and the observed signal ( ). According to [2], we can apply standard LMF algorithm to adaptive channel estimation, with the cost function ( ) = 14 ( ), (2)

where ( ) = ( ) − ( ) ( ) is -th adaptive updated error. The update equation of the filter can be written as ( + 1) = ( ) + ∂ ( )∂ (n) = ( ) + ( ) ( ), (3)

where denotes the step-size of gradient descend. Unfortunately, channel estimation using standard LMF algorithm is not stable in adaptive updating process and hence it cannot be employed directly [2]. To improve the reliability of LMF, an stable LMF algorithms was proposed by virtual of normalization in [3], which was termed as normalized LMF (NLMF) algorithm. The update equation is given by ( + 1) = ( ) + ( ) ( )‖ ( )‖ ‖ ( )‖ + ( )

= ( ) + ( ) ( )‖ ( )‖ , (4)

where = ( )‖ ( )‖ + ( ), (5)

denotes variable step-size of gradient descent and ‖∙‖ is the Euclidean norm operator and ‖ ‖ = ∑ | | .

Unfortunately, interpretation of the relation between

and ( ) is not correct in [7]. Here, we observe that when ( ) ≫ ‖ ( )‖ , then approaches to ; when ( ) ≈‖ ( )‖ , then approaches to 2⁄ ; when ( ) ≪ ‖ ( )‖ , then approaches to 0. Fig. 3 illustrates the intuitive relation between and ( ) . For the standard NLMF algorithm, consider three step-sizes : 0.2, 0.6 and 1.2 . As ( ) increases, according to the depicted curves in Fig. 3, it is easy to find that the stability of NLMF approaches NLMS which stability was proven in [8]. Please also note that when ( )

Fig. 2. A typical example of sparse multipath channel.

Fig. 3. Relations between and ( ).

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III. AD

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here ∈w ; = [stribution atrix; = [ctor which is

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ASCE using Recall that th

andard NLMFethod does notused by its oilize the sparsASSO algoritnstraint to thenction as

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DAPTIVE SPAR

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gorithm channel estimn Eq. (4), howage of the chanfunction in (2

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sparse constraint function to the cost function in (4), adopt different re-weighted factors, e.g., = 1,5,10 and 20, then we are able to penalize different sparse constraint strength on different channel coefficients. In other words, RZA can penalize stronger constraint strength than smaller channel coefficients, and vice versa. Based on this idea, an improved ASCE using RZA-NLMF algorithm is proposed in the following.

C. ASCE using RZA-NLMFalgorithm The ZA-NLMF cannot distinguish between zero taps and

non-zero taps since all the taps are forced to zero uniformly as show in Fig. 5. Unfortunately, ZA-NLMF based approach will degrade amount of estimation performance. Motivated by reweighted ℓ -minimization sparse recovery algorithm [6] in CS [12,13], we proposed an improved ASCE method using RZA-NLMF algorithm. The cost function of RZA-NLMF is given by ( ) = 14 ( ) + log(1 + |ℎ |/ ), (10)

where > 0 is a regularization parameter which trades off the estimation error and channel sparsity. The corresponding update equation is ( + 1) = ( ) + ( ) ( )‖ ( )‖ + sgn ( )1 + | ( )|, (11)

where = / is a parameter which depends on step-size , regularization parameter and threshold , respectively. In

the second term of (11), if magnitudes of ℎ ( ), =0,1, … , − 1 are smaller than 1⁄ , then these small coefficients will be replaced by zeros in high probability.

D. Equivalence between CS-based sparse channel estimation and ASCE

According to LASSO algorithm, we explained the connections between CS-based sparse channel estimation method and ASCE. Here, we further interpretation their equivalence between them. It will be useful to enrich sparse signal processing theory. Let take LASSO and ZA-NLMF for an example. If each row of training matrix in system model (6) is , then by virtual of matrix vectorization, then it can be written as [ , … , , . . , ] . The -th receive signal is obtained as = + ζ . Their corresponding functions of different vectors are listed in Tab. I.

TAB. I. EQUIVALENCE BETWEEN LASSO AND ZA-LNMF.

LASSO ZA-NLMF

Training signal ( )

Received signal ( )

Channel vector = [ , , … , ] = [ℎ , ℎ , … , ℎ ]Sparse constraint ‖ ‖ sign( )

Iterative times = mod( − 1, ) + 1

IV. COMPUTER SIMULATIONS In this section, the proposed ASCE method using RZA-

NLMF algorithm is evaluated. For achieving average

performance, 1000 independent Monte-Carlo runs are adopted. The length of channel vector is set as = 16 and its number of dominant taps is set to = 1 and 4, respectively. Each dominant channel tap follows random Gaussian distribution as (0, σ ) and their positions are randomly allocated within the length of which is subject to E{|| || =1} . The received signal-to-noise ratio (SNR) is defined as 10log ( ⁄ ), where = 1 is the unit transmission power. Here, we set the SNR as 3dB and 5dB in computer simulation. All of the step sizes and regularization parameters are listed in Tab. II. The estimation performance is evaluated by average mean square error (MSE) which is defined by Avergae MSE{ ( )} = E{‖ − ( )‖ }, (12)

where E{∙} denotes the expectation operator, and ( ) are the actual channel vector and its -th iterative adaptive channel estimator, respectively.

TAB. II. SIMULATION PARAMETERS.

Parameters Values

Channel length = 16

No. of nonzero coefficients = 1 and 4

Step-size: 1.0 and 1.5

Regularization parameter: and 1e − 5

Re-weighted factor: 5, 10, 20

Two experiments are considered in this section. In the first

experiment, ASCE methods are evaluated in SNR = 5dB . Regularization parameters, i.e., λ and λ , are set as λ =λ = 1 − 6. Fig. 6-8 shows that proposed method can achieve better estimation performance than ZA-NLMF in different step-sizes, i.e., = 0.5 , 1.0 and 1.5 . When the number of nonzero coefficients is = 1, bigger re-weighted factor can obtain better estimation and vice versa. Please note that both ZA-NLMF and RZA-NLMF algorithms apply ASCE

Fig. 6. Average MSE performance comparisons at SNR = 5dB and step-size = 0.5.

which can achieve better estimation performance for sparser channel. As shown in three figures, when = 1, the depicted curves’ gap between NLMF and either ZA-NLMF or RZA-NLMF larger than the case with = 4 . It is worth mentioning that bigger reweighted factor obtains better estimation performance, e.g., RZA-NLMF using ε = 20 achieves better estimation performance than ε = 10 . It is mainly due to the bigger reweighted factor which can obtain variable sparse constraint for different magnitude of channel coefficients.

To verify the flexibility of the proposed RZA-NLMF

method at different SNR regimes, ASCE methods are evaluated in SNR = 10dB . Also, regularization parameters, i.e., λ and λ , are set to λ = λ = 5e − 8 to improve the

estimation performance. According to Fig. 9, RZA-NLMF works well and achieves better estimation performance than ZA-NLMF and NLMF.

V. CONCLSION In this paper, the disadvantage of conventional ASCE

method using ZA-NLMF algorithm was discussed from a geometrical perspective point of view. We found that ℓ -norm sparse constraint penalizes uniformly on different channel coefficients. Inspired by re-weighted ℓ -norm algorithm in CS, we proposed an improved ASCE using RZA-NLMF which penalizes smaller channel coefficients with stronger sparse constraint and vice versa. It was confirmed by computer simulation that the proposed ASCE using RZA-NLMF algorithm achieves better performance than either ZA-NLMF or NLMF approaches.

ACKNOWLEDGMENT This work was supported by the Japan Society for the

Promotion of Science (JSPS) postdoctoral fellowship and the National Natural Science Foundation of China under Grant 61261048.

REFERENCES

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[5] G. Gui and F. Adachi, “Sparse least mean forth filter with zero-attracting L1-norm constraint,” Submitted for International Conference

Fig. 7. Average MSE performance comparisons at SNR = 5dB and step-size = 1.0.

Fig. 8. Average MSE performance comparisons at SNR = 5dB and step-size = 1.5.

Fig. 9. Average MSE performance comparisons at SNR = 10dB and step-size = 1.5.

on Information, Communications and Signal Processing (ICICS), Tainan, Taiwan, Dec. 10-13, 2013.

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