Design of a class of multirate systems using a maximum relative ℒ/sup 2/-error criterion

Abstract

A criterion for designing the class of multirate systems for rate-changing is presented. This criterion arises from a model-matching perspective with maximum relative < 2 -error over a general class of inputs and is a natural extension of the standard Chebyshev method for lter design. Using multirate and convex analysis techniques, the criterion is shown to lead to a convex matrix-valued-function approximation problem. An algorithm using convex optimization is proposed to solve the problem. An example illustrates the use of the algorithm and eectiveness over standard lter-design techniques.

Optimal Approximate Inverse of Linear Periodic Filters

“…The procedure for constructing one such with minimal possible reconstruction delay is given in [10, p. 198]. b) Formulation of other signal design and reconstruction problems as an optimal model-matching problem is also found in the designs of rate-changing multirate elements [2], [18], and perfect reconstruction filterbank, e.g., see [19] [where the same 2-norm criterion as in (4.4) is used], and [3], [5], [6] (in which an criterion is used). We should note, however, that the problems addressed in all these works do not consider the noise effect.…”

“…For a given reconstruction delay, we show how to construct an approximate inverse, which is a causal and stable periodic filter with the same period, such that the average energy of the block reconstruction error is kept small. There is a natural formulation of the optimization problem in terms of transfer matrices of periodic filters as an optimal model-matching problem [2], [5]. Owing to the noise effect, it is seen that the resultant model and plant are nonsquare rational matrices.…”

“…Define the following two augmented rational matrices: 1)-(4.3), we can express the objective function in the so-called model-matching form[2],[5],[6], as(4.4) …”

Optimal Design of Transform-Based Block Digital Filters Using a Quadratic Criterion

“…To evaluate the quality of a Block Digital Filter, we must define a criterion which measures the distance between the obtained output spectrum and the desired output spectrum. We will use a standard quadratic criterion (mean square error): This is the most widely used criterion, while other, more sophisticated, criteria exist (see [2][3] for instance). In the equation above, f (k) is the desired frequency response.…”

“…From equation 17 we can see that the aliasing at a given frequency k is aliasing(k) = L−1 r=1 p(r, k − br) 2 . This value can be represented as a function of k, as shown in figure 13.…”