Amplify-and-compute: Function computation over layered networks

We study layered wireless networks in which destinations decode linear combinations of transmitted messages over a finite field. We propose amplify-and-compute, a simple but effective scheme in which sources encode their messages with lattice codes, relays employ amplify-and-forward relaying, and destinations decode incoming signals to integer combinations of lattice codewords. We focus on the two-user setting and show that, by carefully choosing relay amplification weights, it is possible to align the equivalent end-to-end channel to an arbitrary matrix of non-zero integers (up to a set of channel matrices of zero measure). Such a choice achieves a computation rate to within a gap to capacity that is independent of the SNR. Amplify-and-compute therefore achieves the maximum degrees of freedom while providing good performance at moderate SNR. Finally, we show that amplify-and-compute offers similar performance when applied to layered two-user interference networks.