Calibration Schemes with $\mathcal{O}(N\log{N})$ Scaling for Large-N
  Radio Interferometers Built on a Regular Grid
  Future generations of radio interferometers targeting the 21\,cm signal at
cosmological distances with $N\gg 1000$ antennas could face a significant
computational challenge in building correlators with the traditional
architecture, whose computational resource requirement scales as
$\mathcal{O}(N^2)$ with array size. The fundamental output of such correlators
is the cross-correlation products of all antenna pairs in the array. The
FFT-correlator architecture reduces the computational resources scaling to
$\mathcal{O}(N\log{N})$ by computing cross-correlation products through a
spatial Fourier transform. However, the output of the FFT-correlator is
meaningful only when the input antenna voltages are gain- and phase-calibrated.
Traditionally, interferometric calibration has used the $\mathcal{O}(N^2)$
cross-correlations produced by a standard correlator. This paper proposes two
real-time calibration schemes that could work in parallel with an
FFT-correlator as a self-contained $\mathcal{O}(N\log{N})$ correlator system
that can be scaled to large-N redundant arrays. We compare the performance and
scalability of these two calibration schemes and find that they result in
antenna gains whose variance decreases as $1/\log{N}$ with increase in the size
of the array.
