Content-Centric Multicast Beamforming in Cache-Enabled Cloud Radio Access Networks

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1 Content-Centric Multicast Beaforing in Cache-Enabled Cloud Radio Access Networks Hao Zhou, Meixia Tao, Erkai Chen, Wei Yu *Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Dept. of Electrical and Coputer Engineering, University of Toronto, Toronto, Canada Eail: {zhouhao zh, xtao, arxiv: v2 [cs.it] 11 Aug 215 Abstract Multicast transission and wireless caching are effective ways of reducing air and backhaul traffic load in wireless networks. This paper proposes to incorporate these two key ideas for content-centric ulticast transission in a cloud radio access network RAN) where ultiple base stations BSs) are connected to a central processor CP) via finite-capacity backhaul links. Each BS has a cache with finite storage size and is equipped with ultiple antennas. The BSs cooperatively transit contents, which are either stored in the local cache or fetched fro the CP, to ultiple users in the network. Users requesting a sae content for a ulticast group and are served by a sae cluster of BSs cooperatively using ulticast beaforing. Assuing fixed cache placeent, this paper investigates the joint design of ulticast beaforing and content-centric BS clustering by forulating an optiization proble of iniizing the total network cost under the quality-of-service QoS) constraints for each ulticast group. The network cost involves both the transission power and the backhaul cost. We odel the backhaul cost using the ixed l /l 2-nor of beaforing vectors. To solve this nonconvex proble, we first approxiate it using the seidefinite relaxation SDR) ethod and concave sooth functions. We then propose a difference of convex functions DC) prograing algorith to obtain suboptial solutions and show the connection of three sooth functions. Siulation results validate the advantage of ulticasting and show the effects of different cache size and caching policies in cloud RAN. I. INTRODUCTION Cloud radio access network RAN) is an eerging network architecture capable of exploiting the advantage of ulticell cooperation in the future fifth-generation 5G) wireless syste [1]. In a cloud RAN, the base stations BSs) are connected to a central processor CP) via digital backhaul links, thus enabling joint data processing and precoding capabilities across ultiple BSs. This paper proposes a content-centric view for cloud RAN design. We equip the BSs with finite-size cache, where popular contents desired by ultiple users can be stored. We forulate and solve a network optiization proble while accounting for the finite-capacity backhaul links between the BSs and the CP. To address the issue of liited backhaul, previous works on wireless cooperative networks [2] [4] consider the proble of iniizing the backhaul traffic and transission power by designing sparse beaforer and user-centric BS clustering. Further, [5] considers the weighted su rate WSR) optiization proble under per-bs backhaul constraints. However, all these works focus on the unicast scenario and proote a usercentric view of syste design without considering the effect of caching. Recently, wireless caching has been investigated as an effective way of reducing peak traffic and backhaul load. By deploying caches at BSs and placing popular contents in the in advance, the issue of liited backhaul capacity can be addressed fundaentally. In [6], the authors show that with sall or even no backhaul capacity, feto-caching can support high deand of wireless video distribution. In [7], the upper and lower bounds of the capacity of the caching syste are derived, and it is shown that the network capacity could be further iproved by using coded ulticasting for content delivery. These studies otivate us to consider the cache-enabled cloud RAN, where each BS is equipped with a cache with finite storage size. Copared with cooperative networks without caching, cache-enabled cloud RAN can fundaentally reduce the backhaul cost and support ore flexible BS clustering. We note that in [8], a siilar wireless caching network has been considered, where the authors study the data assignent and unicast beaforing design. Different fro previous work focused on unicast [2] [4], [8], where data is transitted to each user individually no atter whether the actual contents requested by different users are the sae or not, this paper focuses on the proble of ulticast transission. We assue that ultiple users can request the sae content, and the content is delivered using ulticast beaforing to these users on the sae resource block. Copared with traditional unicast, ulticast can iprove energy and spectral efficiency. In addition, since the popular contents cached in the BSs are possibly requested by ultiple users, ulticast could better exploit the potential of wireless caching. This paper studies the joint design of ulticast beaforing and content-centric BS clustering, which differs fro the fixed BS clustering in coordinated uticell ulticast networks [9] or user-centric BS clustering in unicast systes [3]. In each scheduling interval, the BS clustering is dynaically optiized with respect to each ulticast group. We forulate an optiization proble with the objective of iniizing the total power consuption as well as the backhaul cost under the quality-of-service QoS) constraints for each ulticast group. The backhaul cost is forulated as a function of the ixed l /l 2 -nor of the beaforing vectors. The challenge in solving such a proble is due to both the non-convex QoS constraints and the l -nor in the backhaul cost. In this

2 2 paper, we first use the seidefinite relaxation SDR) ethod introduced in [1] to handle the non-convex QoS constraints. We then adopt the sooth function approach in sparse signal processing to approxiate the l -nor with concave sooth functions. In sparse signal processing, one approach to handle the l -nor iniization proble is to approxiate the l - nor with its reweighted l 1 -nor [11] and update the weight factors iteratively. Another approach is the sooth function ethod [4], where the authors use Gaussian faily functions to approxiate the l -nor. The sooth function ethod is a better approxiation to the l -nor but its perforance highly depends on the soothness factor of the approxiation function. In this paper, we adopt the sooth function approach and solve the approxiated proble with a difference of convex functions DC) algorith [12]. We explore the use of three different sooth functions, the logarithic function, the exponential function, and the arctangent function, and show that with a particular weight updating rule, all three are equivalent to the reweighted l 1 -nor iniization [11]. Siulation results are presented to illustrate the perforance of proposed algorith and the benefit of wireless caching. Notations: Boldface uppercase letters denote atrices and boldface lowercase letters denote colun vectors. The sets of coplex nubers and binary nubers are denoted as C and B respectively. The statistical expectation, transpose and Heritian transpose are denoted as E ), ) T and ) H respectively. The Frobenius nor and the l -nor are denoted as 2 and respectively. An all-one vector of length M is denoted as 1 M. An all-zero vector of length M is denoted as M. The inner product of atrices X and Y is defined as X,Y = TrX H Y ). For a square atrix S M M, S eans that S is positive seidefinite. A. Signal Model II. SYSTEM MODEL We consider a cache-enabled cloud RAN with one CP, L BSs and K obile users. Each BS is equipped with N t antennas and each user has a single antenna. Scheduling and beaforer design are done at the CP. Each BS is connected to the CP via a finite-capacity backhaul link. The total nuber of contents is F ; different contents are independent. The CP stores all the contents and there is a cache at each BS, which stores finite nuber of contents. Each user requests a content according to the content popularity, and users requesting the sae content for a ulticast group. We assue that the total nuber of ulticast groups is M. The set of users in group is denoted as K with M K = K. We study the cooperative downlink ulticast transission and dynaic content-centric BS clustering. Each group is served by a cluster of BSs cooperatively, denoted as Q. In each scheduling interval, the BS clustering {Q } M is dynaically optiized by the CP. For exaple, in Fig. 1, the instantaneous BS clusters for different groups are Q 1 = {1,2,3}, Q 2 = {2} and Q 3 = {2,3}, respectively. For BS 3, since it serves both group 1 and group 3, it should acquire Multicast Group Backhaul Link Data Link Interference Content Provider Central Processor Cache Cache Cache BS 1 BS 2 BS 3 Group 1 Group 2 Group 3 Fig. 1: An exaple of downlink cloud RAN with M = 3 groups and L = 3 cache-enabled BSs connected to a CP via digital backhaul links, where each ulticast group is served by a cluster of BSs. the contents for these two groups either fro its local cache or through backhaul. We denote the aggregate beaforing vector of group fro all BSs as w C LNt 1 = [w1, H,wH 2,,,wH L, ]H, where w l, C Nt 1 is the beaforing vector for group at BS l. Note that the BS clustering is iplicitly defined by the beaforing vectors. If the beaforing vector w l, is Nt, then BS l does not serve group and is thus not in Q. On the other hand, if w l, Nt, BS l is part of the serving cluster of group. Thus, the size of the BS cooperation cluster for group can be expressed as a ixed l /l 2 -nor of the beaforing vector {w l, } L l=1, i.e. Q = L l=1 wl, 2 2. We denote the data sybol of the content requested by group as s C, with E [ s 2] = 1. For user k K, its received downlink signal y k can be written as y k = h H k w s + h H k w ns n +z k, 1) n where h k C LNt 1 is the network-wide channel vector fro all BSs to user k and z k CN,σ 2 ) is the addictive noise. The received signal-to-interference-plus-noise ratio SINR) at user k K is SINR k = h H k w 2 M n hh k w 2) n 2 +σ2. We define the target SINR vector as γ = [γ 1,γ 2,,γ M ] with each eleent γ being the target SINR to be achieved by the users in group. In this paper, we consider the fixed rate transission as in [3], where the transission rate for group is set as R = log 2 1+γ ). Thus, to successfully decode the essage, for any user k K, its achievable data rate should be larger than R, that is, for,k K, log 2 1+SINR k ) R.

3 3 B. Cache Model We assue that each content has noralized size of 1 and the local storage size of BS l is F l F l < F), which is also the axiu nuber of contents it can store. Therefore, we define a cache placeent atrix C B L F, where c l,f = 1 eans the content f is cached in BS l and c l,f = eans the opposite. Note that l, F f=1 c l,f F l. We assue that the cache placeent is static, that is, atrix C is fixed and is known at the CP siilar assuptions have been ade in previous literature, e.g., [8]). This assuption is based on the fact that the optiization of cache placeent is perfored on a large tie scale; while the beaforing design is done on a sall tie scale of channel coherence tie. Hence, it is reasonable to assue that during a short scheduling interval, the cache placeent policy reains unchanged. C. Cost Model We consider the total network cost which consists of both the transission power and the backhaul cost. Let f denote the content requested by users in ulticast group. For BS l in Q, if content f is in its cache, it can access the content directly without costing backhaul. On the contrary, if content f is not cached, BS l needs to first fetch this content fro the CP via the backhaul link. Since the data rates of fetching the contents fro the CP need to be as large as the contentdelivery rate, the backhaul cost in this case is odeled as the transission rates of ulticast groups. The total backhaul cost at all BSs can be written as C B = l=1 wl, 2 2 R 1 c l,f ). 3) The total network cost can be written as C N = η w l=1 wl, 2 2 R 1 c l,f ), }{{}}{{} Power Consuption Backhaul Cost 4) where η is a weight paraeter. By adjusting the value of η, we can ephasize on one cost versus the other. Note that in a network without caching, there is a tradeoff between power and backhaul cost. To reduce power consuption, each group can be served by ore BSs, which increases backhaul cost. However, in cache-enabled cloud RAN, for each group, the BSs caching the requested content can be involved in the cooperative cluster of the group without costing extra backhaul. III. PROBLEM FORMULATION AND APPROXIMATION In this section, we present the optiization proble of iniizing the total network cost by jointly designing ulticast beaforing and BS clustering. We show that this proble is a non-convex proble and further approxiate it with two steps. A. Proble Forulation Our objective is to iniize the total network cost, under the constraints of the peak transission power at each BS and the SINR requireent of each ulticast group. The optiization proble is forulated as P : iniize C N 5a) {w } M subject to SINR k γ,,k K 5b) w l, 2 2 P l, l 5c) where P l is the peak transission power at BS l. Proble P is a non-convex proble, where the nonconvexity coes fro both the l -nor in the objective function and the SINR constraints. Unlike traditional unicast beaforing proble where the non-convex SINR constraints can be transfored to a second-order cone prograing SOCP) proble and the optial solutions can be obtained with convex optiization, the ulticast beaforing proble is NP-hard [1]. In this paper, we use two techniques to approxiate problep, naely SDR relaxation and l -nor approxiation. The overall procedure to solve P is shown in Fig. 2, with each step elaborated in following sections. B. Step 1 SDR Relaxation In both single-cell [1] and ulticell [9] scenarios, the seidefinite relaxation SDR) ethod has been used to deal with the non-convex SINR constraints in ulticast beaforing design probles. We define two sets of atrices {W C LNt LNt } M and {H k C LNt LNt } K k=1 as W = w w H and H k = h k h H k,,k. 6) We further define a set of selecting atrices{ } L l=1, where each atrix B LNt LNt is a diagonal atrix defined as ]) = diag [ H,1 H l 1)Nt Nt, H, l. 7) L l)nt Therefore, we have w l, 2 2 = TrW ), l,. 8) By adopting the SDR ethod, proble P can be relaxed and rewritten as P SDR: iniize {W } M subject to ηtrw )+ l=1 α l, TrW ) TrW H k ) M n TrW nh k )+σ 2 γ,,k K TrW ) P l, l W, 9a) 9b) 9c) 9d)

4 4 where, to further siplify the atheatical representation, we have defined a set of constants {α l, },,M l=1,,l with α l, = R 1 c l,f ). The SINR constraints in proble P SDR are convex. We denote the resulting optial {W } after solving proble P SDR as {W }. If W is already rank-one, then for group the optial aggregate beaforer w of proble P can be obtained by applying eigen-value decoposition to W as W = λ ŵŵ H and taking w = λ ŵ. Otherwise, the beaforing vectors {w } can be generated with the randoization ethod used in [9] and [1]. SDR Approach Sooth Function Approxiation P PSDR PSF w t) Obtain w fro. Initialization t=: Solve Pini : V w ). Iteration t : Update l,, Solve Pt: Vt, w t). Vt-1 Vt < Yes DC Algorith No t = t +1 C. Step 2 l -nor Approxiation To solve the proble P SDR, we further approxiate the non-convex l -nor in the objective with a continuous function denoted as fx). Specifically, we consider three frequently used sooth functions: logarithic function, exponential function and arctangent function [13], defined as TrX)+ ) log, for log-function fx) = 1 e TrX), ) for exp-function 2 π arctan TrX), for atan-function 1) where is a paraeter to adjust the soothness of the functions. In all three cases, with larger, the function is soother but is a worse approxiation to the l -nor. In [4], the authors use the Gaussian faily sooth functions, where the l -nor of w 2 is approxiated with f w 2 ) = 1 exp w ). In this work, by adopting the 2 exponential sooth function, the l -nor is approxiated with f exp W) = 1 exp TrW) ). Coparing f w 2 ) and f exp W), we can see that the exponential sooth function in 1) has the sae approxiation effect as the Gaussian sooth function in [4]. With sooth function, P SDR can be rewritten as P SF : iniize {W } M ηtrw )+ l=1 α l, f W ) 11a) subject to 9b), 9c), 9d). 11b) For ease of presentation, we express the objective as the suation of two functions GW) and FW), defined as GW) = ηtrw ) and FW) = l=1 α l, f l,, 12) where f l, = f W ), l,. We see that GW) and FW) are an affine and a concave function of W, respectively, so proble P SF can be viewed as the difference of two continuous convex functions with convex constraints. Therefore, this proble can be solved with the DC algorith, which falls in the category of ajorization iniization MM) algoriths [12]. Fig. 2: Overall procedure for solving proble P IV. DC BASED ALGORITHM AND ANALYSIS In this section, we first present the DC based algorith for solving the proble P SF using the logarithic sooth function. We then show that the resulting algorith can also be interpreted as a DC algorith with the other two sooth functions and a different updating rule. A. DC Based Algorith with Log-Function The DC algorith iteratively optiizes an approxiated convex function of the original concave objective function and produces a sequence of iproving {W }. The algorith converges to soe global/local optial solution. The initial {W ) } is found by solving the following power iniization proble P ini : V ini iniize {W } M TrW ) 13a) subject to 9b), 9c), 9d). 13b) In the t-th iteration, {W t) } is generated as the solution of the approxiated convex optiization proble, P t : V t iniize GW) {W } M ) + α l, t 1) W f l,, l=1 where W t 1) W W t 1) ) 14a) subject to 9b), 9c), 9d). 14b) f l, is the gradient atrix of f l, at W t 1). Specifically, for log-function, the gradient atrix W f l, C LNt LNt at {W t) } is D log l,) = t) W f l, = ), 15) Tr W t) + If we let the soothness factor = ǫ, where ǫ is a very sall positive constant. Then we have D log l,) =. 16) Tr W t) )+ǫ

5 5 Note that 16) has the siilar for as the weight factor of the reweighted l 1 -nor approach in [11]. Thus, the DC algorith with log-function and soothness factor is just the reweighted l 1 -nor iniization algorith of [3]. In 14a), function FW) is approxiated with its firstorder Taylor expansion, which provides an upper bound. This algorith terinates when the sequence of {W t) } converges to soe stationary point, and the objective value V t converges, that is, V t 1 V t <, where is a sall constant. B. Updating Rule of for Other Sooth Functions The perforance of l -nor approxiation algoriths depends on the soothness factor. Intuitively, when x is large, should be large so that the approxiation algorith can explore the entire paraeter space; when x is sall, should be sall so that fx) has behavior close to l -nor. In [14], the authors propose to use a decreasing sequence of, but the updating rule does not depend on x. In this paper, we explore a novel updating rule that achieves the above effect autoatically using a sequence of which depends on specific x in each iteration. More specifically, we propose to set to be the one that axiizes the gradient of the approxiation function. Note that the gradient atrices of the exponential and arctangent functions in 1) are, respectively, and D exp l,) = e TrW ), 17) D atan l,) = 2 π TrW ) ) 2 +l,. 18) An interesting observation is that for all three functions in 1), if we axiize their gradients, we get expressions of the sae for. Specifically, for the log-function, we get 16) with optial l,. For the exp-function and atan-function, we get D expl,) = ax D exp l,) = Datan l,) = axd atan l,) = e TrW t) ), 19) π TrW t) ), 2) respectively with the optial l, = TrWt) ). We see that the gradient atrices for all three approxiation functions in 1) differ by a constant ultiple only. Therefore, they lead to the sae algorith if we update such that W t) f l, is axiized in the t-th iteration, i.e., l, argax t) W l, f l,. 21) In this proposed algorith, the approxiation functions are adjusted dynaically to achieve a good tradeoff between soothness and approxiation to l -nor. Further, siilarity between 16), 19), and 2) eans that with proposed updating rule, these three approxiation functions lead to alost the sae perforance. V. SIMULATION RESULTS We consider a cache-enabled cloud RAN covering an area of circle with the radius of 1.2k, where 7 BSs L = 7,N t = 3) are placed in a equilateral triangular lattice with the distance between adjacent BSs of.8k. The total nuber of contents is F = 1. A total nuber of 14 users are distributed in this network with unifor distribution and they are scheduled in a round-robin anner. In each scheduling interval, K = 14 users are scheduled. We assue half of the scheduled users request a coon content e.g., a live video) and each of the rest randoly requests one content according to the content popularity, which is odeled as Zipf distribution with skewness paraeter 1. Users requesting the sae content participate in the sae ulticast group. We assue all BSs have the sae cache size. The channels between BSs and users are generated with a noralized Rayleigh fading coponent and a distance-dependent path loss, odeled as P LdB)= log 1 d) with 8dB log-noral shadowing, where d is the distance fro the user to the BS. The transit antenna power gain at each BS is 1 dbi. The power spectral density of downlink noise is 172dB/Hz with the channel bandwidth of 1MHz. The peak transission power of each BS is P l = 1W, l. The target SINR of each content is 1dB. We set = 1 6 for the convergence condition in the DC algorith and ǫ = 1 7 in 16). Each siulation result is averaged over 3 scheduling intervals. In Fig. 3, we show the effects of wireless caching and the cache size. The popularity-aware cache refers to the policy where each BS caches the contents with highest popularity. The figure shows that copared with the network without cache, the cache-enabled network can reduce the backhaul cost by ore than 5% when each BS only caches 5% of the total contents. The backhaul reduction is up to 75% when each BS can cache 3% of the total contents. In Fig. 4, we copare the effects of different cache strategies. In the rando cache policy, the contents are randoly cached with equal probability in each BS. The result shows that the popularity-aware cache has better power-backhaul efficiency in general. In specific, with the sae transission power cost of 38dB, and the sae cache size of 1, popularity-aware cache only costs about 1/4 backhaul cost of the rando cache policy. However, in the extree case when we do not consider the total transission power, the iniu backhaul costs of the two caching strategies are about the sae if the cache size is 3. In Fig. 5, we copare the power-backhaul tradeoff of ulticast and unicast transission in the sae scenario. We use popularity-aware caching policy with the cache size of 1. In unicast transission, users of the sae group are served by different beaforers and we use the algorith proposed in [3], where in each iteration, for k,l, the weight factor ρ l k is set to if the content requested by user k is cached in BS l. If ultiple users in the sae group are served by a sae BS, the backhaul cost is counted only once. The figure shows that in the power-liited syste, the total power consuption

6 6 Total Backhaul bps/hz) η + No Cache Popularity Aware Cache, Size = 5 Popularity Aware Cache, Size = 1 Popularity Aware Cache, Size = 3 η Total Backhaul bps/hz) η + Multicast Transission Unicast Transission η 2 1 Total Backhaul bps/hz) Total Power Consuption db) Fig. 3: Power-backhaul tradeoff for different cache size η + Popularity Aware Cache, Size = 1 Rando Cache, Size = 1 Popularity Aware Cache, Size = 3 Rando Cache, Size = η Total Power Consuption db) Fig. 4: Power-backhaul tradeoff for different cache policies. of ulticast transission is about 2dB less than the unicast scenario. When the total power consuption is 38dB, the backhaul cost of ulticast transission is only 1/3 of unicast transission. These observations validates the advantage of caching and ulticasting in such a scenario. VI. CONCLUSION This paper investigates the joint design of ulticast beaforing and content-centric BS clustering in a cache-enabled cloud RAN. The optiization proble is forulated as the iniization of the total network cost, including the power consuption and the backhaul cost, under the QoS constraint of each ulticast group. We adopt the SDR ethod and the sooth function approach, introduced in sparse signal processing, to approxiate this non-convex proble as a DC prograing proble. We then propose a DC algorith to obtain sub-optial solutions. Further, we propose a new soothness updating ethod and give insight into its connection to reweighted l 1 -nor iniization. Siulation results show that, copared with unicast transission, ulticast transission can achieve better power-backhaul tradeoff, and Total Power Consuption db) Fig. 5: Power-backhaul tradeoff for ulticast and unicast transission. the backhaul cost can be further reduced with larger cache size. REFERENCES [1] P. Rost, C. Bernardos, A. Doenico, M. Girolao, M. Lala, A. Maeder, D. Sabella, and D. Wubben, Cloud technologies for flexible 5G radio access networks, IEEE Coun. Mag., vol. 52, no. 5, pp , May 214. [2] J. Zhao, T. Q. Quek, and Z. Lei, Coordinated ultipoint transission with liited backhaul data transfer, IEEE Trans. Wireless Coun., vol. 12, no. 6, pp , 213. [3] B. Dai and W. Yu, Sparse beaforing for liited-backhaul network MIMO syste via reweighted power iniization, in Proc. IEEE Global Telecoun. Conf, 213. [4] F. Zhuang and V. Lau, Backhaul liited asyetric cooperation for MIMO cellular networks via seidefinite relaxation, IEEE Trans. Signal Process., vol. 62, no. 3, pp , Feb 214. [5] B. Dai and W. Yu, Sparse beaforing and user-centric clustering for downlink cloud radio access network, IEEE Access, vol. 2, pp , 214. [6] N. Golrezaei, A. Molisch, A. Diakis, and G. Caire, Fetocaching and device-to-device collaboration: A new architecture for wireless video distribution, IEEE Coun. Mag., vol. 51, no. 4, pp , April 213. [7] M. Maddah-Ali and U. Niesen, Fundaental liits of caching, IEEE Trans. on Infor. Theory, vol. 6, no. 5, pp , May 214. [8] X. Peng, J.-C. Shen, J. Zhang, and K. B. Letaief, Joint data assignent and beaforing for backhaul liited caching networks, International Syposiu on Personal, Indoor and Mobile Radio Counications PIMRC), 214. [9] Z. Xiang, M. Tao, and X. Wang, Coordinated ulticast beaforing in ulticell networks, IEEE Trans. Wireless Coun., vol. 12, no. 1, pp , January 213. [1] E. Karipidis, N. D. Sidiropoulos, and Z.-Q. Luo, Quality of service and ax-in fair transit beaforing to ultiple cochannel ulticast groups, IEEE Trans. Signal Process., vol. 56, no. 3, pp , 28. [11] E. J. Candes, M. B. Wakin, and S. P. Boyd, Enhancing sparsity by reweighted l 1 iniization, Journal of Fourier analysis and applications, vol. 14, no. 5-6, pp , 28. [12] R. Horst and N. V. Thoai, DC prograing: overview, Journal of Optiization Theory and Applications, vol. 13, no. 1, pp. 1 43, [13] F. Rinaldi, F. Schoen, and M. Sciandrone, Concave prograing for iniizing the zero-nor over polyhedral sets, Coputational Optiization and Applications, vol. 46, no. 3, pp , 21. [14] H. Mohiani, M. Babaie-Zadeh, and C. Jutten, A fast approach for overcoplete sparse decoposition based on soothed l nor, IEEE Trans. Signal Process., vol. 57, no. 1, pp , 29.