Degrees of freedom of two-hop wireless networks: “Everyone gets the entire cake”

We show that fully connected two-hop wireless networks with K sources, K relays and K destinations have K degrees of freedom for almost all values of constant channel coefficients. Our main contribution is a new interference-alignment-based achievability scheme which we call aligned network diagonalization. This scheme allows the data streams transmitted by the sources to undergo a diagonal linear transformation from the sources to the destinations, thus being received free of interference by their intended destination.