3:30, SPTM-P11.1
THE DESIGN OF EQUIRIPPLE MATRIX FILTERS
C. MACINNES
This paper presents a procedure for deriving linear filters which are based on matrix-vector multiplication instead of linear convolution and which can be designed to match the frequency responses of linear equiripple FIR filters. The magnitude of the matrix filter response is matched to the magnitude response of a given linear FIR filter by solving a set of nonlinear equations numerically using Broyden's method.

3:30, SPTM-P11.2
ALL-PASS DIGITAL FILTER DESIGN IN THE FREQUENCY-DELAY DOMAIN USING THE ITERATIVE QUADRATIC MAXIMUM LIKELIHOOD ALGORITHM
J. TAPIA, G. ATKIN, J. LOCICERO
All-Pass Digital Filter Design in the Frequency-Delay Domain Using the Iterative Quadratic Maximum Likelihood Algorithm Abstract - A new domain, termed the frequency-delay domain, is used to design stable, all-pass digital filters resembling a given delay response in the least-squares sense. This spectral technique identifies the delay response of a stable, second-order, all-pass digital filter as a double sideband suppressed carrier amplitude modulated signal in the frequency-delay domain. Iterative Maximum Likelihood techniques are used to render the filter coefficients. The algorithm is a significant improvement over related methods because it results in a physically realizable stable all-pass filter that closely approximates a desired delay response.

3:30, SPTM-P11.3
UNIFIED DESIGN ALGORITHM FOR COMPLEX FIR AND IIR FILTERS
W. LERTNIPHONPHUN, J. MCCLELLAN
In this paper, a general filter design norm is proposed with the intent of producing a unified design algorithm for all types of filters---FIR, IIR and 2-D FIR with complex specifications. The Chebyshev, least squares, and constrained least squares problems become special cases because this norm uses a convex combination of the 2-norm and the Chebyshev norm. The primary benefit of this new problem formulation is that a single efficient multiple exchange algorithm (similar to Remez) has been developed to cover all the different filter types for magnitude and phase approximation. In the new algorithm, a small subproblem is formed at each step and is solved with an iterative reweighted least squares technique which can handle the design of complex filters easily. Finally, the norm definition allows easy trade-offs between the relative importance of error energy and worst-case error.

3:30, SPTM-P11.4
OPTIMAL DESIGN METHOD FOR FIR FILTER WITH DISCRETE COEFFICIENTS BASED ON INTEGER SEMI-INFINITE LINEAR PROGRAMS
R. ITO, K. SUYAMA, R. HIRABAYASHI
The purpose of the paper is to propose a new design method of FIR filters with discrete coefficients considering optimality. In the proposed method, the design problem of FIR filters is formulated as a Mixed Integer Semi-Infinite Linear Programming problem (MISILP), which can be solved by a branch and bound technique. Then, it is possible to obtain the optimal discrete coefficients, and the optimality of the obtained solution can be guaranteed. It was confirmed that optimal coefficients of linear phase FIR filter with discrete coefficients could be designed in reasonable computational time with sufficient precision based on the results of computational experiments.

3:30, SPTM-P11.5
FAST ALGORITHM FOR LEAST SQUARES 2D LINEAR-PHASE FIR FILTER DESIGN
N. GRISWOLD, J. DAVILA
In this paper, we develop a new method for weighted least squares 2D linear-phase FIR filter design. It poses the problem of filter design as the problem of projecting the desired frequency response onto the subspace spanned by an appropriate orthonormal basis We show how to compute the orthonormal basis efficiently in the cases of quadrantally-symmetric filter design and centro-symmetric filter design. The design examples show that the proposed method is faster than a conventional weighted least squares filter design mthod. Also, the amount of storage required to compute the filter coefficients is greatly reduced.

3:30, SPTM-P11.6
LINEAR MATRIX INEQUALITY FORMULATION OF SPECTRAL MASK CONSTRAINTS
Z. LUO, J. STURM, T. DAVIDSON
The design of a finite impulse response filter often involves a spectral `mask' which the magnitude spectrum must satisfy. This constraint can be awkward because it yields an infinite number of inequality constraints (two for each frequency point). In current practice, spectral masks are often approximated by discretization, but in this paper we will show that piecewise constant masks can be precisely enforced in a finite and convex manner via linear matrix inequalities. This facilitates the formulation of a diverse class of filter and beamformer design problems as semidefinite programmes. These optimization problems can be efficiently solved using recently developed interior point methods. Our results can be considered as extensions to the well-known Positive-Real and Bounded-Real Lemmas from the systems and control literature.

3:30, SPTM-P11.7
DESIGN OF LINEAR PHASE IIR FILTERS VIA WEIGHTED LEAST-SQUARES APPROXIMATION
P. AGATHOKLIS, C. XIAO, J. OLIVER
A new method for designing IIR digital filters with linear phase in the passband is proposed. This method is based on frequency weighted least-square error optimization using the BFGS method \cite{iir:fle}. The gradient of the cost function with respect to the design parameters, required for the implementation of the BFGS method, is derived. The proposed method is started by obtaining an initial IIR filter design using model reduction of a linear phase FIR filter. Based on this initial design the cost function is minimized using the BFGS method. An example shows that the proposed method leads to very good filter designs.

3:30, SPTM-P11.8
INTERIOR-POINT METHODS FOR MAGNITUDE FILTER DESIGN
B. ALKIRE, L. VANDENBERGHE
We describe efficient interior-point methods for the design of filters with constraints on the magnitude spectrum, for example, piecewise-constant upper and lower bounds, and arbitrary phase. Several researchers have observed that problems of this type can be solved via convex optimization and spectral factorization. The associated optimization problems are usually solved via linear programming or, more recently, semidefinite programming. The semidefinite programming approach is more accurate but also more expensive, because it requires the introduction of a large number of auxiliary variables. In this paper we propose a more efficient method, based on convex optimization duality, and on interior-point methods for problems with generalized inequalities.

3:30, SPTM-P11.9
SDP DESIGN PROCEDURE FOR ENERGY COMPACTION IIR FILTERS
R. NIEMISTÖ, B. DUMITRESCU, I. TABUS
In this paper we present a design method for optimal energy compaction IIR filters, where the numerator and denominator may have different degrees. The design is performed via iterative relaxations, where the numerator is optimized given the denominator, followed by optimization of denominator given the numerator. The two optimization problems involved are solved using semi-definite programming (SDP) techniques, where the real positiveness of the causal part of the product filter is formulated in two alternative ways: first using Kalman-Yakubovich-Popov (KYP) lemma, and second, by a less known parameterization, which we show to be more convenient numerically. Numerical results show the effectiveness of the proposed method and the improvements when compared with optimal FIR compaction filters or constrained IIR compaction filters (restricted to have allpass polyphase components).

3:30, SPTM-P11.10
ON THE DESIGN OF LP IIR FILTERS WITH ARBITRARY FREQUENCY RESPONSE
R. VARGAS, C. BURRUS
This paper introduces an iterative algorithm for designing IIR digital filters that minimize a complex approximation error in an $L_p$ sense. The algorithm combines ideas that have proven successful in the similar problem of $L_p$ FIR filter design. We use iterative prefiltering techniques common in applications such as parameter estimation together with an Iterative Reweighted Least Squares (IRLS) method. The result is a double iterative approach that generates IIR filters of arbitrary magnitude and phase response and arbitrary numerator and denominator orders. Such filters can be used in a variety of applications in which the typical $L_2$ or $L_\infty$ error criteria might not be suitable.

3:30, SPTM-P11.11
ALL-PASS FILTER DESIGN USING PROJECTION-BASED METHOD UNDER GROUP DELAY CONSTRAINTS
K. HADDAD, Y. YANG, N. GALATSANOS, H. STARK
A new technique for designing digital all-pass IIR filters is proposed. The approach is based on the vector space projection method. Constraint sets, and their associated projectors, that capture the properties of the desired group delay are given. Examples that demonstrate the advantages and flexibility of this method as well as comparisons with a well- known method are furnished.