The role of linear semi-infinite programming in signal-adapted QMF bank design

Abstract

The design of an orthogonal FIR quadrature{mirror lter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a di cult nonlinear constrained optimization problem. In this paper, it is shown that in fact the coding gain depends only upon the product lter P (z) = H (z)H(z ?1), and this transformation leads to a stable class of linear optimization problems having nitely many variables and in nitely many constraints, termed linear semi{ in nite programming (SIP) problems. The sought{for, original lter, H (z), is obtained by de ation and spectral factorization of P (z). With the SIP formulation, every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms. The natural regularity properties inherent in the SIP formulation enhance the performance of these algorithms. We present a comprehensive theoretical analysis of the SIP problem and its dual, characterize the optimal lters, and analyze uniqueness and sensitivity issues. All these properties are intimately related to those of the input signal and bring considerable insight into the nature of the adaptation process. We present discretization and cutting plane algorithms and apply both methods to several examples.

Theory and Design of Optimum Fir Compaction Filters,

“…6. From the plot, we see that the best compaction gain is for V. LINEAR PROGRAMMING METHOD The use of linear programming (LP) method in compaction filter design was proposed by Moulin et al [20], [31], [32]. We briefly review the method and propose some improvements.…”

Optimal waveform design for UWB radios

“…However, (14b) defines an infinite number of constraints that must be rendered finite. One way to approximate (14b) is to sample it uniformly in frequency and replace it by for all (15) Typically is chosen to be on the order of . The parameter can be chosen such that satisfaction of (15) guarantees satisfaction of (14b) even for (e.g., [26]), or it can be chosen to be a smaller number.…”

“…This problem is a semi-infinite linear program (SILP; e.g., [15]) in which there are two constraints for each . The linear constraint set in (16c) is a necessary and sufficient condition for to represent a valid autocorrelation sequence.…”

“…The linear constraint set in (16c) is a necessary and sufficient condition for to represent a valid autocorrelation sequence. The semi-infinite constraints can be discretized to form a finite linear program whose solution can be efficiently found [15], [16]. While that solution is an approximation to the true solution, discretization strategies exist that ensure this approximation to be a good one; e.g., the direct analogy of (15).…”

“…The semi-infinite constraints can be discretized to form a finite linear program whose solution can be efficiently found [15], [16]. While that solution is an approximation to the true solution, discretization strategies exist that ensure this approximation to be a good one; e.g., the direct analogy of (15). As an alternative to discretization, the constraint in (16c) can be precisely enforced using a single linear matrix inequality (LMI) constraint (e.g., [10]), which will result in a (convex) semidefinite programming (SDP; [6], [23]) formulation of the design problem (e.g., [9]- [11]).…”