arXiv subjects

Haichao Wang

Publications and source records attributed to Haichao Wang.

21 records · Page 2Linked to original sources

arXiv2012

Uncertainty Principles in Finitely generated Shift-Invariant Spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in $L^2(\R^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in $L^2(\R)$ whose generator and its translates form a Riesz basis.

math.FA↗

arXiv2010

Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces

In this paper, we consider the time-frequency localization of the generator of a principal shift-invariant space on the real line which has additional shift-invariance. We prove that if a principal shift-invariant space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any $n\ge2$, there exist principal shift-invariant spaces on the real line that are also $\nZ$-invariant with an integrable orthonormal (or a Riesz) generator $ϕ$, but $ϕ$ satisfies $\int_{\mathbb R} |ϕ(x)|^2 |x|^{1+ε} dx=\infty$ for any $ε>0$ and its Fourier transform $\hatϕ$ cannot decay as fast as $ (1+|ξ|)^{-r}$ for any $r>1/2$. Examples are constructed to demonstrate that the above decay properties for the orthormal generator in the time domain and in the frequency domain are optimal.

math.FA↗

arXiv2009

Sequential adaptive compressed sampling via Huffman codes

There are two main approaches in compressed sensing: the geometric approach and the combinatorial approach. In this paper we introduce an information theoretic approach and use results from the theory of Huffman codes to construct a sequence of binary sampling vectors to determine a sparse signal. Unlike other approaches, our approach is adaptive in the sense that each sampling vector depends on the previous sample. The number of measurements we need for a k-sparse vector in n-dimensional space is no more than O(k log n) and the reconstruction is O(k).

cs.IT↗