Research Article An Improved Proportionate Normalized Least-Mean-Square Algorithm for Broadband Multipath Channel Estimation

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1 e Scientific World Journal, Article ID 72969, 9 ages htt://dx.doi.org/1.11/214/72969 Research Article An Imroved Proortionate Normalized Least-Mean-Square Algorithm for Broadband Multiath Channel Estimation Yingsong Li and Masanori Hamamura Graduate School of Engineering, Kochi University of Technology, Kami-shi , Jaan Corresondence should be addressed to Yingsong Li; liyingsong@ieee.org Received 27 December 213; Acceted 19 February 214; Published 2 March 214 Academic Editors: H. R. Karimi, X. Yang, Z. Yu, and W. Zhang Coyright 214 Y. Li and M. Hamamura. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. To make use of the sarsity roerty of broadband multiath wireless communication channels, we mathematically roose an l - norm-constrained roortionate normalized least-mean-square (-) sarse channel estimation algorithm. A general l - norm is weighted by the gain matrix and is incororated into the cost function of the roortionate normalized least-mean-square () algorithm. This integration is equivalent to adding a zero attractor to the iterations, by which the convergence seed and steady-state erformance of the inactive tas are significantly imroved. Our simulation results demonstrate that the roosed algorithm can effectively imrove the estimation erformance of the -based algorithm for sarse channel estimation alications. 1. Introduction Broadband signal transmission is becoming a commonly used high-data-rate technique for next-generation wireless communication systems, such as 3 GPP long-term evolution (LTE) and worldwide interoerability for microwave access (WiMAX) [1]. The transmission erformance of coherent detection for such broadband communication systems strongly deends on the quality of channel estimation [2 ]. Fortunately, broadband multiath channels can be accurately estimated using adative filter techniques [6 1] suchas the normalized least-mean-square () algorithm, which has low comlexity and can be easily imlemented at the receiver. On the other hand, channel measurements have shown that broadband wireless multiath channels can often be described by only a small number of roagation aths with long delays [4, 11, 12]. Thus, a broadband multiath channel can be regarded as a sarse channel with only a few active dominant tas, while the other inactive tas are zero or close to zero. This inherent sarsity of the channel imulse resonse (CIR) can be exloited to imrove the quality of channel estimation. However, such classical algorithms with a uniform ste size across all filter coefficients have slow convergence when estimating sarse imulseresonsesignalssuchasthoseinbroadbandsarse wireless multiath channels [11]. Consequently, corresonding algorithms have recently received significant attention in the context of comressed sensing (CS) [, 12 14] and were already considered for channel estimation rior to the CS era [, 12]. However, these CS channel estimation algorithms are sensitivetothenoiseinwirelessmultiathchannels. Insired by the CS theory [12 14], several zero-attracting (ZA) algorithms have been roosed and investigated by combining the CS theory and the standard least-meansquare (LMS) algorithm for echo cancellation and system identification, which are known as the zero-attracting LMS (ZA-LMS) and reweighted ZA-LMS (RZA-LMS) algorithms, resectively [1]. Recently, this technique has been exanded to the algorithm and other adative filter algorithms to imrove their convergence seed in a sarse environment [9, 16 18]. However, these aroaches are mainly designed for nonroortionate adative algorithms. On the other hand, to utilize the advantages of the algorithm, such as stable erformance and low comlexity, the roortionate

2 2 The Scientific World Journal normalized least-mean-square () algorithm has been roosedandstudiedtoexloitthesarsityinnature[19]and has been alied to echo cancellation in telehone networks. Although the algorithm can utilize the sarsity characteristics of a sarse signal and obtain faster convergence at the initial stage by assigning indeendent magnitudes to the active tas, the convergence seed is reduced by even more than that of the algorithm for the inactive tas after theactivetasconverge.consequently,severalalgorithms have been roosed to imrove the convergence seed of the algorithm [2 27], which include the use of the l 1 -norm technique and a variable ste size. Although thesealgorithmshavesignificantlyimrovedtheconvergence seed of the algorithm, they still converge slowly after the active tas converge. In addition, some of them are inferior to the and algorithms in terms of the steady-state error when the sarsity decreases. From these reviously roosed sarse signal estimation algorithms, we know that the ZA algorithms mainly exert a enalty on the inactive channel tas through the integration of the l 1 - norm constraint into the cost function of the standard LMS algorithms to achieve better estimation erformance, while the algorithm udates each filter coefficient with an indeendent ste size, which imroves the convergence of the active tas. Motivated by the CS theory [13, 14] andzatechnique [1 18], we roose an l -norm-constrained (- ) algorithm that incororates the l -norm into the cost function of the algorithm, resulting in an imroved roortionate adative algorithm. The difference between the roosed - algorithm and the ZA algorithms is that the gain-matrix-weighted l -norm is used in our roosed - algorithm instead of the general l 1 -norm to exand the alication of ZA algorithms [1]. Also, this integration is equivalent to adding a zero attractor in the iterations of the algorithm to obtain the benefits of both the and ZA algorithms. Thus, our roosed - algorithm can achieve fast convergence attheinitialstagefortheactivetas.aftertheconvergence of these active tas, the ZA technique in the - algorithm acts as another force to attract the inactive tas to zero to arrest the slow convergence of the algorithm. Furthermore, our roosed - algorithm achieves a lower mean square error than the algorithm and its related imroved algorithms, such as the imroved (I) [2] andμ-law (M) [21] algorithms. In this study, our roosed - algorithm is verified over a sarse multiath channel by comarison with the,, I, and M algorithms. The simulation results demonstrate that the - algorithm achieves better channel estimation erformance in terms of both convergence seed and steady-state behavior for sarse channel estimation. The remainder of this aer is organized as follows. Section 2 briefly reviews the standard,, andimrovedalgorithms,includingthei and M algorithms. In Section 3, we describe in detail the roosed - algorithm, which emloys the Lagrange multilier method. In Section4, the estimation erformance of the roosed - algorithm is verified over sarse channels and comared with other commonly used algorithms. Finally, this aer is concluded in Section. 2. Related Channel Estimation Algorithms 2.1. Normalized Least-Mean-Square Algorithm. In this section, we first consider the sarse multiath communication system shown in Figure 1 to discuss the channel estimation algorithms. The inut signal x(n)=[x(n),x(n 1),...,x(n N+1)] T containing the N most recent samles is transmitted over a finite imulse resonse (FIR) channel with channel imulse resonse (CIR) h = [h,h 1,...,h N 1 ] T,where( ) T denotes the transosition oeration. Then the outut signal of the channel is written as follows: y (n) = h T x (n), (1) where h is a sarse channel vector with K dominant active taswhosemagnitudesarelargerthanzeroand(n K) inactive tas whose magnitudes are zero or close to zero with K N.Toestimatetheunknownsarsechannelh,an algorithm uses the inut signal x(n), the outut signal y(n), and the instantaneous estimation error e(n), which is given by e (n) =d(n) ĥt (n) x (n), (2) where ĥ(n) istheadativechannelestimatoratinstant n, d(n) = y(n) + V(n),andV(n) is an additive noise at the receiver. The udate function of the channel estimation algorithm is exressed as ĥ (n+1) = ĥ (n) +μ e (n) x (n) x T, (3) (n) x (n) +δ where μ is the ste size with <μ <2and δ is a small ositive constant used to avoid division by zero Proortionate Normalized Least-Mean-Square Algorithm. The algorithm, which is an algorithm imroved by the use of a roortionate technique, has been roosed for sarse system identification and echo cancellation. In this algorithm, each ta is assigned an individual ste size, which is obtained from the revious estimation of the filter coefficient. According to the gain allocation rule in this algorithm, the greater the magnitude of the ta, the larger the ste size assigned to it, and hence the active tas converge quickly. The udate function of the algorithm [19] is described by the following equation with reference to Figure 1: ĥ (n+1) = ĥ (n) +μ e (n) G (n) x (n) x T. (n) G (n) x (n) +δ (4) Here, G(n), which denotes as the gain matrix, is a diagonal matrix that modifies the ste size of each ta, μ is the

3 The Scientific World Journal 3 Inut signal x(n) Unknown FIR channel h Estimated FIR channel ĥ(n) Adative algorithms Adative channel estimation Additive noise Outut signal + (n) y(n) + ŷ(n) e(n) + d(n) Figure 1: Tyical sarse multiath communication system. global ste size of the algorithm, and δ =δ 2 x /N is a regularization arameter to revent division by zero at the initialization stage, where δ 2 x is the ower of the inut signal x(n). In the algorithm, the gain matrix G(n) is given by G (n) = diag (g (n),g 1 (n),...,g N 1 (n)), () where the individual gain g i (n) is defined as with g i (n) = γ i (n) N 1 i= γ i (n), i N 1 (6) γ i (n) = max [ρ g max [δ, ĥ (n), ĥ1 (n),..., ĥn 1 (n) ], ĥi (n) ], (7) where the arameters δ and ρ g are ositive constants with tyical values of δ =.1 and ρ g = /N. δ is used to regularize the udating at the initial stage when all the tas areinitializedtozero,andρ g is used to revent ĥi(n) from stalling when it is much smaller than the largest coefficient Imroved Proortionate Normalized Least-Mean-Square Algorithms I Algorithm. The I algorithm is a tye of algorithm used to imrove the convergence seed of the algorithm. It is a combination of the and algorithms with the relative significance of each coefficient controlled by a factor α. The I algorithm [2] adotsthel 1 -norm to enable the smooth selection of (7), and the udate equation of the I algorithm is exressed as ĥ (n+1) = ĥ (n) μ e (n) K (n) x (n) I x T, (n) K (n) x (n) +δ I (8) where K(n) = diag(k (n), k 1 (n),...,k N 1 (n)) is a diagonal matrix used to adjust the ste size of the I algorithm, where k j (n) = 1 α 2N + (1+α) (n) ĥj 2 ĥ (n) +ε, 1 j N 1 for a small ositive constant ε and 1 α 1.Attheinitial stage,thestesizeismultiliedby(1 α)/2n, sinceallthe filter coefficients are initialized to zero. Thus, in the I algorithm, a regularization arameter δ I is introduced, which is given by (9) δ I = 1 α 2N δ. (1) We can see that the I is identical to the algorithmforα = 1, while the I behaves identically to the algorithm when α=1. In ractical engineering alications, a suitable value for α is or M Algorithm. The μ-law algorithm (M) is another enhancement of the algorithm that utilizes the logarithm of the magnitudes of the filter coefficients instead of using the magnitudes directly in the algorithm [21]. The udate equation is the same as that in the algorithm given by (4). In the M algorithm, γ i (n) = max [ρ g max [δ F( ĥ (n) ),F( ĥ1 (n) ),..., where F( ĥn 1 (n) )], F( ĥi (n) )], (11) F( (n) ĥi )=log (1 + θ (n) ĥi ), (12) where θ is a large ositive constant related to the estimation accuracy requirement, tyically θ = Proosed - Algorithm In this section, we roose an - algorithm by incororating the l -norm into the cost function of the algorithm to create a zero attractor, making it a tye of ZA algorithm. The difference between the - algorithm and general ZA algorithms is that the gain-matrixweighted l -norm is taken into account in designing the zero attractor. On the other hand, the roosed - algorithm is based on the commonly used algorithm, which is also a sarse channel estimation algorithm and can imrove the convergence for the active tas. Regarding

4 4 The Scientific World Journal channel estimation, the urose of the - algorithm is to minimize (ĥ (n+1) ĥ (n))t G 1 (n) where ε is a small value to revent division by zero. By multilying both sides of (17)byx T (n),weobtain x T (n) ĥ (n+1) = xt (n) ĥ (n) +λxt (n) G (n) x (n) (ĥ (n+1) ĥ (n))+γ G 1 (n)ĥ(n + 1) subject to d (n) ĥt (n+1) x (n) =, (13) 1 x T (n) ĥ(n) γ. 1 ĥ(n) +ε (18) where G 1 (n) is the inverse of the gain matrix G(n) in the algorithm, γ >is a very small constant used to balance the estimation error and the sarse l -norm enalty of ĥ(n + 1), is the -norm defined as ĥ =( i ĥ i )1/, and 1.Notethatin(13), we introduce an l -norm enalty to ĥ(n + 1) after scaling the gain matrix by G 1 (n), which is different from the reviously roosed ZA LMS algorithms. To minimize (13),theLagrangemultiliermethodis adoted, and the cost function J (n + 1) of the roosed - algorithm is exressed as J (n+1) = (ĥ (n+1) ĥ (n))t G 1 (n) (ĥ (n+1) ĥ (n))+γ G 1 ĥ(n + 1) +λ (d (n) ĥt (n+1) x (n)), (14) where λ isthelagrangemultilier. By calculating the gradient of the cost function J (n + 1) of the - algorithm and assuming ĥ(n + 1) = ĥ(n) in the steady stage, we have J (n+1) ĥ (n+1) =, J (n+1) λ =, (1) ĥ (n+1) = ĥ (n) +λg (n) x (n) γ ĥ(n). 1 ĥ(n) (16) 1 In ractice, we need to introduce a small ositive constant into the final term in (16) tocoewiththesituationthatan entry of ĥ(n) aroaches zero, which is the case for a sarse CIR at initialization. Then the udate equation (16)ofthe- algorithm is modified to ĥ (n+1) = ĥ (n) +λg (n) x (n) γ ĥ(n), 1 ĥ(n) +ε (17) 1 From (2), (1), and (17), we obtain e (n) = γ x T (n) ĥ(n) 1 ĥ(n) 1 +ε +λx T (n) G (n) x (n). (19) Then, the Lagrange multilier λ is given as follows by solving (19): 1 x T (n) ĥ(n) λ= (e(n) +γ ) 1 ĥ(n) +ε (x T (n) G (n) x (n)) 1. Substituting (2)into(17), we have 1 ĥ (n+1) = ĥ (n) γ ĥ(n) 1 ĥ(n) +ε = 1 (2) x T (n) ĥ(n) + ((e (n) +γ ) 1 ĥ(n) +ε (x T (n)g(n)x(n)) 1 ) G (n) x (n) ĥ (n) + e (n) G (n) x (n) x T (n) G (n) x (n) γ {I G (n) x (n) xt (n) x T (n) G (n) x (n) } 1 ĥ (n). ĥ (n) 1 +ε (21) It was found that the magnitudes of the elements in the matrix G(n)x(n)x T (n){x T (n)g(n)x(n)} 1 are much smaller

5 The Scientific World Journal than 1 for broadband multiath channel estimation. Therefore, the udate equation (21) of the roosed - algorithm is rewritten as ĥ (n+1) = ĥ (n) + e (n) G (n) x (n) x T (n) G (n) x (n) 1 γ ĥ(n). 1 ĥ(n) +ε (22) Here, we neglect the effects of the matrix G(n)x(n)x T (n) {x T (n)g(n)x(n)} 1 and assume that the filter order is large. Similarly to the algorithm, a ste size μ is introduced to balance the convergence seed and the steady-state error of the roosed - algorithm, and a small ositive constant ε =δ 2 x /N is emloyed to revent division byzero.thus,theudatefunction(22)canbemodifiedto ĥ (n+1) = ĥ (n) +μ e (n) G (n) x (n) x T (n) G (n) x (n) +ε ρ ĥ(n) 1 ĥ(n) 1 +ε = ĥ (n) +μ e (n) G (n) x (n) x T ρ (n) G (n) x (n) +ε T (n), (23) where ρ =μ γ and T(n)= ĥ(n) 1 sgn(ĥ(n)){ ĥ(n) 1 + ε } 1.Comaringtheudatefunction(23) oftheroosed - algorithm with the udate function (4) ofthe algorithm, we see that our roosed - algorithm has the additional term γ T(n), also defined as the zero attractor, which attracts the small channel tas to zero with high robability. Moreover, the ZA strength of this zero attractor is controlled by ρ. In other words, in our roosed - algorithm, the gain matrix G(n) assigns a large stesizetotheactivechanneltasofthesarsechannel,while the zero attractor mainly exerts the l -enalty on the inactive tas whose tas are zero or close to zero. Thus, our roosed - algorithm can further imrove the convergence seed of the algorithm after the convergence of the large active tas. 4. Results and Discussions In this section, we resent the results of comuter simulations carried out to illustrate the channel estimation erformance oftheroosed-algorithmoverasarsemultiath communication channel and comare it with those of the reviously roosed I, M,, and algorithms. Here, we consider a sarse channel h whose length N is 64 or 128 and whose number of dominant active tas K is set to three different sarsity levels, namely, K=2,4and 8, similar to revious studies [6, 22, 2, 26]. The dominant active channel tas are obtained from a Gaussian Magnitude Channel ta index Figure 2: Tyical sarse multiath channel. distribution with h 2 2 =1, and the ositions of the dominant channel tas are randomly saced along the length of the channel. The inut signal x(n) of the channel is a Gaussian random signal while the outut of the channel is corruted by an indeendent white Gaussian noise V(n). An examle ofatyicalsarsemultiathchannelwithachannellength of N = 64 and a sarsity level of K = 3 is shown in Figure 2.Inthesimulations,theowerofthereceivedsignal is E b =1,whilethenoiseowerisgivenbyδ 2 V and the signal -to-noise ratio is defined as SNR = 1 log(e b /δ 2 V ).Inallthe simulations, the difference between the actual and estimated channels based on the sarsity-aware algorithms and the sarse channel mentioned above is evaluated by the MSE defined as follows: MSE (n) =1log 1 E { h ĥ(n) } (db). (24) 2 In these simulations, the simulation arameters are chosen to be μ =μ =μ I =μ =., δ =.1, ε =.1, α=, ε =., ρ =1 1, δ =.1, ρ g =/N, θ = 1, =., andsnr=3db.whenwe change one of these arameters, the other arameters remain constant Estimation Performance of the Proosed - Algorithm Effects of Parameters on the Proosed - Algorithm. Intheroosed-algorithm,therearetwo extra arameters, and ρ,comaredwiththe algorithm, which are introduced to design the zero attractor. Next, we show how these two arameters affect the roosed - algorithm over a sarse channel with N=64or 128 and K = 4. The simulation results for different values of ρ and are shown in Figures 3 and 4, resectively. From Figure 3(a),wecanseethatthesteady-stateerrorofthe - algorithm decreases with decreasing ρ when 2

6 6 The Scientific World Journal ρ =4 1 ρ =3 1 ρ =2 1 ρ =1 1 ρ =2 1 6 ρ =2 1 7 ρ =4 1 ρ =3 1 ρ =2 1 ρ =1 1 ρ =2 1 6 ρ =2 1 7 (a) N=64 (b) N = 128 Figure 3: Effects of ρ on the roosed - algorithm. ρ 2 1 6, while it increases again when ρ is less than Furthermore, the convergence seed of the - algorithm raidly decreases when ρ is less than 1 1. This is because a small ρ results in a low ZA strength, which consequently reduces the convergence seed. In the case of N = 128 shown in Figure 3(b),weobservethat both the convergence seed and the steady-state erformance are imroved with decreasing ρ for ρ 1 1.When ρ < 1 1,theconvergenceseedofthe- algorithm decreases while the steady-state error remains constant. Figure4 demonstrates the effects of the arameter. We can see from Figure 4(a) that the convergence seed of the roosed - algorithm raidly decreases with increasing for N = 64. Moreover, the steady-state error is reduced with ranging from.4 to., while it remains constant for =.6,.7, and.8. However, the steadystate erformance for =1isinferior to that for =.8. Thisisbecausetheroosed-algorithmisan l 1 -norm-enalized algorithm, which cannot distinguish between active tas and inactive tas, reducing its convergence seed and steady-state erformance. When N= 128, asshowninfigure 4(b), the steady-state erformance is imroved as increases from.4 to.6. Thus, we should carefully select the arameters ρ and to balance the convergence seed and steady-state erformance for the roosed - algorithm EffectsofSarsityLevelontheProosed-Algorithm. On the basis of the results discussed in Section for our roosed - algorithm, we choose =. and ρ = 1 1 to evaluate the channel estimation erformance of the - algorithm over a sarse channel with different channel lengths of N=64and 128, for which the obtained simulation results are given in Figures and 6, resectively.fromfigure, weseethatour roosed - algorithm has the same convergence seed as the algorithm at the initial stage. The roosed - algorithm converges faster than the algorithm as well as the I and algorithms for all sarsity levels K, while its convergence is slightly slower than that of the M algorithm before it reaches a steady stage. However, the roosed - algorithm has the smallest steady-state error for N = 64. When N = 128, we see from Figure 6 that our roosed - algorithm not only has the highest convergence seedbutalsoossessesthebeststeady-stateerformance. This is because with increasing sarsity, our roosed - algorithm attracts the inactive tas to zero quickly and hence the convergence seed is significantly imroved, while the reviously roosed algorithms mainly adjust the ste size of the active tas and thus they only imact on the convergence seed at the early iteration stage. Additionally, we see from Figures and 6 that both theconvergenceseedandthesteady-stateerformanceof all the algorithms deteriorate when the sarsity level K increases for both N = 64 and 128. Inarticular, when K = 8, the convergence seeds of the and I algorithms are greater than that of the algorithm at the early iteration stage, while after this fast initial convergence, their convergence seeds decrease to less than that of the algorithm before reaching a steady

7 The Scientific World Journal =.4 =. =.6 =.7 =.8 =1 =.4 =. =.6 =.7 =.8 =1 (a) N=64 (b) N = 128 Figure 4: Effects of on the roosed - algorithm. stage. Furthermore, we observe that the M algorithm is sensitive to the length N of the channel, and its convergence seed for N = 128 is less than that for N=64at the same sarsity level K andlessthanthatoftheroosed- algorithm. Thus, we conclude that our roosed - algorithm is suerior to the reviously roosed algorithms in terms of both the convergence seed and the steady-state erformance with the aroriate selection of the related arameters and ρ.fromtheabovediscussion, we believe that the gain-matrix-weighted l -norm method inthe-algorithmcanbeusedtofurtherimrove the channel estimation erformance of the I and M algorithms Comutational Comlexity. Finally, we discuss the comutational comlexity of the roosed - algorithm and comare it with those of the,, I, and M algorithms. Here, the comutational comlexity is the arithmetic comlexity, which includes additions, multilications, and divisions. The comutational comlexities of the roosed - algorithm and the related and algorithms are shown in Table 1. From Table 1, we see that the comutational comlexity of our roosed - algorithm is slightly higher than those of the M and algorithms, which is due to the calculation of the gradient of the l -norm. Furthermore, the M algorithm has an additional logarithm oeration, which increases its comlexity but is not included in Table 1. However, the - algorithm noticeably increases the convergence seed and significantly Table 1: Comutational comlexity. Algorithms Additions Multilications Divisions 3N 3N N+3 6N+3 N+2 I 4N + 7 N + N + 2 M N+3 7N+3 N+3-4N+4 9N+4 2N+2 imroves the steady-state erformance of the algorithm. In addition, it also has a higher convergence seed and lower steady-state error than the I and M algorithms when the channel length is large.. Conclusion In this aer, we have roosed an - algorithm to exloit the sarsity of broadband multiath channels and to imroveboththeconvergenceseedandsteady-stateerformance of the algorithm. This algorithm was mainly develoed by incororating the gain-matrix-weighted l - norm into the cost function of the algorithm, which significantly imroves its convergence seed and steady-state erformance. The simulation results demonstrated that our roosed - algorithm, which has an accetable increase in comutational comlexity, increases the convergenceseedandreducesthesteady-stateerrorcomaredwith the reviously roosed algorithms.

8 8 The Scientific World Journal I M - I M - I M - I M - (a) K=2 (b) K=4 (a) K=2 (b) K= I M - (c) K=8 Figure : Effects of sarsity on the roosed - algorithm for N= I M - (c) K=8 Figure 6: Effects of sarsity on the roosed - algorithm for N = 128.

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