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.