Adaptive Identification of Sparse Systems with Variable Sparsity Bijit Kumar Das 1 , Mrityunjoy Chakraborty 2 and Soumitro Banerjee 3 1,2 Department of Electronics and Electrical Communication Engineering Indian Institute of Technology, Kharagpur, INDIA 3 Department of Physical Sciences, IISER, Kolkata, INDIA E.Mail : 1 bijitbijit@gmail.com, 2 mrityun@ece.iitkgp.ernet.in, 3 soumitro.banerjee@gmail.com Abstract— In the context of system identification, it is shown that sometimes the level of sparseness in the system impulse response can vary greatly depending on the time-varying nature of the system. When the response is strongly sparse, convergence of the conventional approach such as least mean square (LMS) is poor. The recently proposed, compressive sensing based sparsity- aware ZA-LMS algorithm performs satisfactorily in strongly sparse environments, but is shown to perform worse than the conventional LMS when sparseness of the impulse response reduces. We propose an algorithm which works well both in sparse and non-sparse circumstances and adapts dynamically to the level of sparseness, using a convex combination based approach. The proposed algorithm is supported by simulation results that show its robustness against variable sparsity. Index terms–Sparse Systems, System Identification, Excess Mean Square Error, Adaptive Filter, l 1 Norm. I. I NTRODUCTION In practice, one often comes across systems that have a sparse impulse response. Further, the degree of sparsity also varies with time and context. Conventional system identifica- tion algorithms like the LMS based adaptive filter [1], however, do not make use of the a priori knowledge of the system sparsity and thus perform poorly both in terms of complexity and convergence speed. In recent years, several algorithms have been proposed that exploit the sparsity of the system and achieve better performance, like the partial update LMS deploying either statistical detection of active taps [2]-[3] or sequential partial updating [4], the proportionate normalized LMS (PNLMS) and its variants [5]-[6] etc. More recently, motivated by LASSO [7] and the recent pro- gresses in compressive sensing [8]-[9], an alternative approach has been proposed in [10] to identify sparse systems which introduces a l 1 norm (of the coefficients) penalty in the cost function which favors sparsity. This results in a modified LMS update with a zero attractor for all the taps, named as the Zero-Attracting LMS (ZA-LMS). A variant of the ZA-LMS is also proposed in [10] which employs reweighted step sizes for the different taps to adjust to variable sparsity. This is, however, associated with huge computational burden due to the L division operations at each step, where L is the number of taps in the filter. Separately, in [11], a class of sparseness- controlled algorithms (SC-PNLMS and SC-MPNLMS) have been proposed, which are robust against the variation of sparsity in the system model while requiring again much higher computational complexity. In this paper, we propose an alternative method to deal with variable sparseness using an adaptive convex combination of a LMS based adaptive filter and a ZA-LMS based adaptive filter. The derivation largely follows the approach of [12] and shows that while for a non-sparse system, the proposed combined filter will always converge to the LMS based adaptive filter, for a sparse system, it will either converge to the ZA-LMS based filter or to a combination that produces lesser excess mean square error (EMSE) than produced by each component filter separately. The proposed algorithm requires much less complexity than required by existing algorithms and its robust- ness against variable sparsity is well supported by simulation results. II. PROBLEM FORMULATION,PROPOSED ALGORITHM AND PERFORMANCE ANALYSIS We consider the problem of identifying a system that takes x(n) as input and produces the observable output y d (n) as y d (n)= w T opt x(n)+ e opt (n), where x(n)=[x(n),x(n - 1), ··· ,x(n - L + 1)] T , w opt is a L × 1 impulse response vector which is known a priori to be sparse and e opt (n) is an observation noise which is assumed to be white with variance σ 2 v and independent of the input data vector x(m) for any m and n. In order to identify the system, we follow the approach of [12] and deploy a convex combination of two adaptive filters as shown in Fig. 1, where Filter 1 uses the ZA-LMS algorithm [10] to adapt a filter coefficient vector w 1 (n) as, w 1 (n +1) = w 1 (n) - ρ sign(w 1 (n)) + μe 1 (n) x(n) (1) and Filter 2 uses the standard LMS algorithm that adapts a filter coefficient vector w 2 (n) as, w 2 (n +1) = w 2 (n)+ μe 2 (n) x(n) (2) where μ is the usual step size (common for both the filters), ρ is a suitable constant and e i (n)= y d (n) - y i (n),i =1, 2 is the respective filter output errors with y i (n)= w T i (n)x(n) denoting the respective filter outputs. The convex combination generates a combined output y(n) = λ(n)y 1 (n) + [1 - λ(n)]y 2 (n). The variable λ(n) is a mixing scalar parameter that lies between 0 and 1, which is to be adapted by following a gradient descent method to minimize the quadratic error function of the overall filter, namely e 2 (n) where e(n) = y d (n) - y(n). However, such adaptation does not guarantee 978-1-4244-9474-3/11/$26.00 ©2011 IEEE 1267