  Radio telescopes that employ arrays of many antennas are in operation, and
ever larger ones are being designed and proposed. Signals from the antennas are
combined by cross-correlation. For $N$ antennas, the cost and power consumption
of cross-correlation are proportional to $N^2$ and dominate at sufficiently
large $N$. Here we report the design of an integrated circuit (IC) that
performs digital cross-correlations for arbitrarily many antennas in a
power-efficient way. It uses an intrinsically low-power architecture in which
the movement of data between devices is minimized. In our design, the
correlations are performed in an array of 4096 complex multiply-accumulate
(CMAC) units. This is sufficient to perform all correlations in parallel for 64
signals ($N$=32 antennas with 2 opposite-polarization signals per antenna).
When $N$ is larger, the input data are buffered in an on-chip memory and the
CMACs are re-used as many times as needed to compute all correlations. The
design has been synthesized and simulated so as to obtain accurate estimates of
the IC's size and power consumption. It is intended for fabrication in a 32 nm
silicon-on-insulator process, where it will require less than 12 mm$^2$ of
silicon area and achieve an energy efficiency of 1.76 to 3.3 pJ per CMAC
operation, depending on the number of antennas. Operation has been analyzed in
detail up to $N=4096$. The system-level energy efficiency, including
board-level I/O, power supplies, and controls, is expected to be 5 to 7 pJ per
CMAC operation.
