Feedback communication using orthogonal signals

This research is concerned with block coding for a feedback communication system in which the forward and feedback channels are independently disturbed by additive white Gaussian noise and average power constrained. Two coding schemes are proposed in which the messages to be coded for transmission over the forward channel are realized as a set of orthogonal waveforms. A finite number of forward and feedback transmissions (iterations) per message is made. Information received over the feedback channel is used to modify the waveform transmitted on successive forward iterations in such a way that the expected value of forward signal energy is zero on all iterations after the first. Similarly, information is sent over the feedback channel in such a way that the expected value of feedback signal energy is also zero on all iterations after the first. These schemes are shown to achieve a lower probability of error than the best one-way coding scheme at all rates up to the forward channel capacity, provided only that the feedback channel capacity be greater than the forward channel capacity. These schemes make more efficient use of the available feedback power than existing feedback coding schemes, and therefore require less feedback power to achieve a given error performance.

Three-dimensional self-assembly of brush block copolymers to photonic crystals

The development of Ring Opening Metathesis Polymerization has allowed the world of block copolymers to expand into brush block copolymers. Brush block copolymers consist of a polymer backbone with polymeric side chains, forcing the backbone to hold a stretched conformation and giving it a worm-like shape. These brush block copolymers have a number of advantages over tradition block copolymers, including faster self-assembly behavior, larger domain sizes, and much less entanglement. This makes them an ideal candidate in the development of a bottom-up approach to forming photonic crystals. Photonic crystals are periodic nanostructures that transmit and reflect only certain wavelengths of light, forming a band gap. These are used in a number of coatings and other optical uses. One and two dimensional photonic crystals are commercially available, though are often expensive and difficult to manufacture. Previous work has focused on the creation of one dimensional photonic crystals from brush block copolymers. In this thesis, I will focus on the synthesis and characterization of asymmetric brush block copolymers for self-assembly into two and three dimensional photonic crystals. Three series of brush block copolymers were made and characterized by Gel Permeation Chromatography and Nuclear Magnetic Resonance spectroscopy. They were then made into films through compressive thermal annealing and characterized by UV-Vis Spectroscopy and Scanning Electron Microscopy. Evidence of non-lamellar structures were seen, indicating the first reported creation of two or three dimensional photonic crystals from brush block copolymers.

Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

The global optimization of phase-incoherent signals

The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_i, i = 1, 2, …, M, on the unit sphere S₁ in C^N. If W_ik is the halfspace determined by ṡ_i and ṡ_k and containing ṡ_i, i.e. W_ik = {ṙϵC^N:| ≥ | ˂ṙ, ṡ_k˃|}, then the Ʀ_i = ∩/k≠i W_ik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S₁. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_ie^jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_C = 1/π^N ∞/ʃ/0 r^2N-1e^-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_i ʃ/∩ S₁ I₀(2r | ˂ṡ, ṡ_i˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/C_α I₀(2r|˂ṡ, ṡ_i˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_α)-σ(S₁)]), where C_α = {ṡϵS₁:|˂ṡ, ṡ_i˃| ≥ α}, K is the total number of boundaries of the net on S₁ determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_α∩W), (where W is a halfspace not containing ṡ_i), given by h = ʃ/C_α∩W I₀ (2r|˂ṡ, ṡ_i˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.