  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 large dish structures. However, the
computational- and storage-costs of such radio arrays, determined by the
$\mathcal{O}(N^2)$ scaling of visibility products required 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. The signal
processing pipeline built for the Hydrogen Epoch of Reionization Array (HERA)
is outlined in this thesis, which implements both fringe stopping and baseline
dependent averaging to bring down the data rate from nearly 1 Tbps to 15 Gbps.
