Browsing by Subject "Spectral mask"

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Open Access
FIR filter design by convex optimization using directed iterative rank refinement algorithm
(Institute of Electrical and Electronics Engineers Inc., 2016) Dedeoğlu, M.; Alp, Y. K.; Arıkan, Orhan
The advances in convex optimization techniques have offered new formulations of design with improved control over the performance of FIR filters. By using lifting techniques, the design of a length-L FIR filter can be formulated as a convex semidefinite program (SDP) in terms of an L× L matrix that must be rank-1. Although this formulation provides means for introducing highly flexible design constraints on the magnitude and phase responses of the filter, convex solvers implementing interior point methods almost never provide a rank-1 solution matrix. To obtain a rank-1 solution, we propose a novel Directed Iterative Rank Refinement (DIRR) algorithm, where at each iteration a matrix is obtained by solving a convex optimization problem. The semidefinite cost function of that convex optimization problem favors a solution matrix whose dominant singular vector is on a direction determined in the previous iterations. Analytically it is shown that the DIRR iterations provide monotonic improvement, and the global optimum is a fixed point of the iterations. Over a set of design examples it is illustrated that the DIRR requires only a few iterations to converge to an approximately rank-1 solution matrix. The effectiveness of the proposed method and its flexibility are also demonstrated for the cases where in addition to the magnitude constraints, the constraints on the phase and group delay of filter are placed on the designed filter.
Open Access
Fir filter design by convex optimization using rank refinement
(2014) Dedeoğlu, Mehmet
Finite impulse response ﬁlters have been one of the primary topics of digital signal processing since their inception. Consequently, diverse class of design techniques including Chebyshev approximation, Fast Fourier Transform and optimization based methods had been proposed in the literature. With developments in com- putational tools, new design technique tools and formulations on ﬁlters including interior-point solvers and semideﬁnite programming (SDP), emerged. Since FIR ﬁlter design problem can be modelled as a quadratically constrained quadratic program, ﬁlter design problem can be solved via interior-point based convex op- timization methods such as semideﬁnite programming. Unfortunately, SDP for- mulation of problem is nonconvex due to positive lower limit constraint in the passband. To overcome that problem, nonconvex problem can be cast into a convex SDP using semideﬁnite relaxation, which can be solved in polynomial time. Since relaxed formulation does not guarantee rank-1 solution matrix, re- cently proposed directed iterative rank reﬁnement (DIRR) algorithm is used to impose a convex rank-1 constraint. Due to utilization of semideﬁnite relaxation and DIRR, addition of various constraints, such as phase and group delay masks, in convex manner is made possible. For feasibility type optimization formulations of ﬁlter design problem, a convergence rate improved version of DIRR is devel- oped. Proposed techniques are applied on ﬁlter design problems with diﬀerent set of constraints including phase and group delay constraints. Explicit simulations demostrate that the proposed technique is capable of solving nonlinear phase, phase constrained, and group delay constrained ﬁlter design problems.