Somantika Datta

Construction of infinite unimodular sequences with zero autocorrelation

Unimodular waveforms x are constructed on the integers with the property that the autocorrelation of x is one at the origin and zero elsewhere. There are three different constructions: exponentials of the form

Efficient Deterministic Compressed Sensing for Images with Chirps and Reed-Muller Codes

e^{2 \pi i n^\alpha \theta},

Construction of k-angle tight frames

sequences taken from roots of unity, and sequences constructed from the elements of real Hadamard matrices. The first is expected and elementary and the second is based on the construction of Wiener. The third is the most intricate and is really one of a family of distinct but structurally similar waveforms. A natural error estimate problem is posed for the last construction. The analytic solution is not as useful as the simulations because of the inherent counting problems in the construction.

Low coherence unit norm tight frames

A recent approach to compressed sensing using deterministic sensing matrices formed from discrete frequency-modulated chirps or from Reed-Muller codes is extended to support efficient deterministic reconstruction of signals that are much less sparse than envisioned in the original work. In particular, this allows the application of this approach in imaging. The reconstruction algorithm developed for images incorporates several new elements to improve computational complexity and reconstruction fidelity in this application regime.

Frame properties of low autocorrelation random sequences

ABSTRACT Frames have become standard tools in signal processing due to their robustness against transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they do exist. However, it is known that an ETF of d + 1 vectors in a d dimensional space always exists. This article gives an explicit construction of ETFs of d + 1 vectors in a d dimensional space. This construction works for both real and complex cases and is simpler than existing methods. The absence of ETFs of arbitrary sizes in a given space leads to generalizations of ETFs. One way to do this to consider tight frames where the set of (acute) angles between pairs of vectors has k distinct values. This article presents a construction of tight frames such that for a given value of k, the angles between pairs of vectors take at most k distinct values. These tight frames can be related to regular graphs and association schemes.

Constructions and a Generalization of Perfect Autocorrelation Sequences on ℤ

Abstract Equiangular tight frames (ETFs) have found significant applications in signal processing and coding theory due to their robustness to noise and transmission losses. ETFs are characterized by the fact that the coherence between any two distinct vectors is equal to the Welch bound. This guarantees that the maximum coherence between pairs of vectors is minimized. Despite their usefulness and widespread applications, ETFs of a given size N are only guaranteed to exist in or if . This leads to the problem of finding approximations of ETFs of N vectors in or where To be more precise, one wishes to construct a unit norm tight frame (UNTF) such that the maximum coherence between distinct vectors of this frame is as close to the Welch bound as possible. In this paper, low coherence UNTFs in are constructed by adding a strategically chosen set of vectors called an optimal set to an existing ETF of vectors. In order to do so, combinatorial objects called block designs are used. Estimates are provided on the maximum coherence between distinct vectors of this low coherence UNTF. It is shown that for certain block designs, the constructed UNTF attains the smallest possible maximum coherence between pairs of vectors among all UNTFs containing the starting ETF of vectors. This is particularly desirable if there does not exist a set of the same size for which the Welch bound is attained.

Finite Frame Theory

Abstract The goal is to construct random frames and study properties of such frames. Starting with the construction of unimodular random sequences whose expected autocorrelations can be made arbitrarily low outside the origin, these random sequences are used to construct frames for ℂd. Using recent theory of non-asymptotic analysis of random matrices, the eigenvalue distribution of the corresponding frame operator is studied.

Welch bounds for cross correlation of subspaces and generalizations

Low autocorrelation signals have fundamental applications in radar and communications. We construct constant amplitude zero autocorrelation (CAZAC) sequences x on the integers ℤ by means of Hadamard matrices. We then generalize this approach to construct unimodular sequences x on ℤ whose autocorrelations A x are building blocks for all functions on ℤ. As such, algebraic relations between A x and A y become relevant. We provide conditions for the validity of the formulas A x+y =A x +A y .

Tight frames, partial isometries, and signal reconstruction

Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely represent any signal in a given space. A frame can be thought of as a redundant basis, having more elements than needed. In fact, in any finite dimensional vector space every finite spanning set is a frame. The redundancy of a frame leads to a non-unique representation, however, this makes signal representation resilient to noise and robust to transmission losses. Frames are now standard tools in signal processing and are of great interest to mathematicians and engineers alike. This chapter presents a brief introduction to frames in finite dimensional spaces, and in particular discusses a highly desirable class of frames called equiangular tight frames. Possible research ideas suitable for an undergraduate curriculum are also discussed.

Sampling of homogeneous polynomials and approximating multivariate functions

Lower bounds on the maximal cross correlation between vectors in a set were first given by Welch and then studied by several others. In this work, this is extended to obtaining lower bounds on the maximal cross correlation between subspaces of a given Hilbert space. Two different notions of cross correlation among spaces have been considered. The study of such bounds is done in terms of fusion frames, including generalized fusion frames. In addition, results on the expectation of the cross correlation among random vectors have been obtained.