The Epoch of Reionization is one of the last unexplored eras of our Universe's history. Beginning about a billion years after the Big Bang, this epoch is characterized by the births of the first stars and galaxies, whose light subsequently altered the nature of the gas surrounding them. There are several experiments aiming to detect the phase transition of this gas as it changes from neutral hydrogen to ionized hydrogen. Such a detection would open a wealth of information about our early Universe, revealing details about the nature of the first luminous sources and the evolution of structure formation.

Interferometers such as the Precision Array for Probing the Epoch of Reionization (PAPER) and the Hydrogen Epoch of Reionization Array (HERA) seek to measure the 21 cm signal from neutral hydrogen and map its evolution over spatial and temporal scales. A detection of reionization, however, is a difficult measurement. Though the 21 cm signal is a powerful 3D probe of the intergalactic medium, it is easily buried underneath bright foreground signals and instrumental systematics. A clean detection of reionization is ambitious and requires analysis methods that maximize data sensitivity and increase confidence in results.

The work presented in this thesis addresses many of the key challenges that face the current field of 21 cm cosmology. This includes algorithms to locate contaminated data, methods to ensure accurate power spectrum measurements, and techniques for removing unwanted systematics while preserving the reionization signal. These developments serve as the foundation of the latest 21 cm measurements from the PAPER-64 and PAPER-128 arrays, whose results lie near the forefront of the field. These methods are also fundamental to HERA and future experiments, as they provide a strong foundation for the continued exploration of our cosmic dawn.

The path towards detecting the cosmological 21 cm signal from the Epoch of Reionization and Cosmic Dawn has seen tremendous progress over the past decade: interferometric experiments have placed increasingly stringent upper-limits on the 21 cm power spectrum, and global signal experiments have made a tentative first-detection of the 21 cm monopole signal. As next-generation experiments are designed, built, and begin taking data, the road towards maximizing their scientific potential hinges crucially upon the development of robust data analysis techniques for isolating the 21 cm signal from foregrounds and low-level instrumental systematics. This thesis is a compilation of work that describes new and old techniques and applies them to the up-and-coming Hydrogen Epoch of Reionization Array (HERA), a second-generation low-frequency experiment that aims to make a detection of the 21 cm power spectrum from redshifts 6 < z < 20. Such a feat would dramatically improve our understanding of the intergalactic medium at these redshifts, and would shed light on the formation and properties of the first stars, galaxies, and compact objects in the universe. Towards this goal, this work touches on some of the current challenges facing the analysis of 21 cm data, from instrument commissioning and calibration to astrophysical parameter inference. In particular, we develop a detailed understanding of low-level systematics in the HERA system, yielding improved sensitivity in the measured power spectrum by over two orders of magnitude. As HERA construction is finished and full-sensitivity observing commences, the algorithms and analysis frameworks discussed here are setting the stage for a future full-sensitivity HERA analysis and the next-generation of 21 cm cosmology.

In the past two decades, a rebirth of interest in low-frequency radio astronomy, for 21\,cm tomography of the Epoch of Reionization, has given rise to a new class of radio interferometers with $N \gg 100$ antennas. The availability of low-noise receivers that do not require cryogenic cooling has driven down the cost of antennas, making it affordable to build sensitivity with numerous small antennas rather than traditional large dish structures. However, the computational- and storage-costs of such radio arrays, determined by the $\mathcal{O}(N^2)$ scaling of visibility products that need to be computed for calibration and imaging, become proportional to the cost of the array itself and drive up the overall cost of the radio telescope.

When antennas in the array are built on a regular grid, direct-imaging methods based on spatial Fourier transforms of the array can be exploited to avoid computing the intermediate visibility matrices that drive the unfavorable scaling. However, such methods rely on the availability of calibrated antenna voltages which are themselves difficult to obtain without using visibility matrices. In this thesis, I explore two real-time calibration strategies that can operate on subsets of visibility matrices, which can be computed without compromising on the $\mathcal{O}(N\log{N})$ scaling of direct-imaging systems.

For more general radio interferometer layouts, baseline-dependent averaging with fringe stopping can be used to decrease the data rate of visibility products. While the computational cost is nearly unchanged, this technique can decrease the data volume of cross-correlation products, making it more tenable to store, process, and calibrate the output of the correlator. In this thesis, I describe the entire signal processing pipeline built for the Hydrogen Epoch of Reionization Array (HERA), which is currently being commissioned for detecting and characterizing the power spectrum of neutral hydrogen in the redshift range $5 < z < 28$. The HERA correlator implements both fringe stopping and baseline dependent averaging to bring down the data rate from nearly 1 Tbps to 15 Gbps.

As one of the last unobserved frontiers in the Universe, the Epoch of

Reionization (EoR) marks the period when the Universe transitioned from a neutral state to an

ionized state, marking the last global phase change. Understanding how the EoR

evolved over time and when it occurred will provide evidence for the nature of

the first luminous sources of the Universe. Specifically, we'll be able to

answer questions such as what were the first galaxies like in terms of their

luminosities, masses, and spectral energy distributions? Were they similar to

todays galaxies? How did they form? How were they spatially distributed? These

questions and many others begin to probe the early times of galaxy formation, a

field in need of observations of the earliest galaxies.

In the first chapter, I provide an introduction to 21 cm cosmology, which uses

the 21 cm line transition from neutral hydrogen to study the evolution of the

Universe. I briefly review 21 cm physics and the evolution of the

cosmological 21 cm signal. The challenges of experiments measuring highly

redshifted neutral hydrogen are discussed before describing the instruments

used to make the measurements of the cosmological 21 cm signal.

The second chapter in this thesis begins by presenting results from Donald C.

Backer Precision Array for Probing the Epoch of Reionization (PAPER) 64 antenna

configuration. I start by describing the dataset used in the chapter, followed by the novel

redundant calibration technique used to calibrate the grid layout of the PAPER array.

I then discuss the application of delay and fringe-rate filtering to suppress foregrounds and

optimally combine time samples, respectively. Before presenting results, I

layout a quadratic estimator formalism for measuring the 21 cm power spectrum.

I finally present the results from the PAPER-64 dataset.

The third chapter of this thesis shifts focus to the new Hydrogen Epoch of

Reionization Array (HERA), a close packed, redundant, low frequency array

dedicated to detecting and characterizing the EoR and Dark Ages. As a new

experiment with terabytes of data being collected per day, the need for a real

time processing system is necessary to avoid a backlog of data and to avoid

data storage issues. I present the real time system (RTS) developed for

analyzing HERA data, which automatically detects and flags bad antennas,

redundantly calibrates the array, and excises radio frequency interference

(RFI) in real time. This work builds off the lessons learned from PAPER and

the Murchison Wide-Field Array (MWA) based in western Australia. I then present

some initial results from the RTS system.

In the final chapter, I conclude the thesis by summarizing the work presented and discuss

future directions for 21 cm cosmology.