A Computable Fourier Condition Generating Alias-Free Sampling Lattices

Abstract

Abstract-We propose a Fourier analytical condition linking alias-free sampling with the Fourier transform of the indicator function defined on the given frequency support. Our discussions center around how to develop practical computation algorithms based on the proposed analytical condition. We address several issues along this line, including the derivation of simple closed-form expressions for the Fourier transforms of the indicator functions defined on arbitrary polygonal and polyhedral domains; a complete and nonredundant enumeration of all quantized sampling lattices via the Hermite normal forms of integer matrices; and a quantitative analysis of the approximation of the original infinite Fourier condition by using finite computations. Combining these results, we propose a computational testing procedure that can efficiently search for the optimal alias-free sampling lattices for a given polygonal or polyhedral shaped frequency domain. Several examples are presented to show the potential of the proposed algorithm in multidimensional filter bank design, as well as in applications involving the design of efficient sampling patterns for multidimensional bandlimited signals.

Sampling trajectories for mobile sensing

“…For ⊂ R d , we use E ⊂ C to denote the collection of all trajectory sets in C that can be expressed in the form (14). The following theorem provides sufficient conditions on the vectors {v 1 , v 2 , .…”

Multidimensional Filter Banks and Multiscale Geometric Representations

“…where δ[n] is the Kronecker delta function (see [63] for a proof). In particular, when n = 0, the above equality implies that…”

“…For the general d-D case, one can show that the total number of such matrices is essentially equal to O(δ d−1 ) [63]. Another useful tool in analyzing MD multirate operations is the Smith normal form [88]: Any integer matrix M can be decomposed into a product U DV , where U and V are unimodular integer matrices and D is an integer diagonal matrix [95].…”

“…A closedform formula for the integral in (5.17) is available when K is an arbitrary polygonal region [63].…”

“…In constructing the 3D DFB, all these desirable traits can be retained, except for one -critically sampling. As much as we would like to have a critically-sampled (that is, nonredundant) 3D DFB, this turns out to be theoretically impossible: In fact, applying the condition (3.11), we can show that, in 3D and beyond, cone-shaped spectrum supports [such as the one in Figure 6.7(a)] can never be critically sampled (see [63,Theorem 3] for a proof). In what follows, we describe the construction of a tree-structured 3D DFB with low redundancy.…”