For any real number θ let where [x] denotes the greatest integer not exceeding x. A method is given for computing fθ from its first few terms. A similar method is given for computing the characteristic function gθ(n) of [nθ]. The given methods converge rapidly, and generalize previous results of Bernoulli, Markorf, and Stolarsky. Note that either of the sequences fθ and gθ determines the sequence [nθ] (n = 1, 2,…).

Determination of [nθ] by its Sequence of*Differences

Further characterizations and properties of exactly covering congruences

We prove that the number of natural exact covering systems of cardinality k is equal to the coefficient of $$x^k$$
 in the reversion of the power series $$\sum _{k \ge 1} \mu (k) x^k$$
 , where 
 $$\mu (k)$$
 is the usual number-theoretic Mobius function. Using this result, we deduce an asymptotic expression for the number of such systems.

Natural exact covering systems and the reversion of the Möbius series

An algebraic characterization of nonuniform perfect reconstruction (PR) filterbanks with integer decimation factors is presented. The PR property is formulated in the z domain based on the response of the linear multirate systems to the delayed unit-step signals. This leads to a unique class of characterizing formulas that are necessary and sufficient conditions for the PR property and free from the complex roots of unity. Two related characterizations of nonuniform PR systems, in the form of necessary conditions, are also developed based on these formulas. As a concrete example, the results are then used to derive necessary and sufficient conditions for PR nonuniform delay chain filterbanks. The conditions show that nonuniform delay chain filterbanks are the signal processing realizations of the mathematical notion of the exact covering systems of congruence relations. Important results from the mathematics literature on the exact covering systems are introduced. The results elucidate the admissible factors of decimation for the nonuniform PR delay chain systems in settings with maximally distinct decimation factors. A simple test of the PR property for delay chain systems is also presented. The test is based on the divisibility of certain polynomials by the cyclotomic polynomials. Finally, multirate systems based on the Beatty sequences, which are the irrational generalization of the exact covering systems, are briefly discussed.

Characterization of nonuniform perfect-reconstruction filterbanks using unit-step signal

The first few problems in this miscellaneous section are about lattice points, whose Euclidean coordinates are integers. Most of them are two-dimensional problems, but some can be formulated in higher dimensions as well.

Tefan Znám

Papers

A simple characterization of disjoint covering systems