Capacity of a Gaussian FIR linear relay network

The conceptual framework for a linear network code for a discrete time relay network with two network modes, additive white Gaussian noise, single antenna nodes, and time invariant flat fading channels and conditions for stable and finite impulse response is presented here. For fixed FIR linear network codes the capacity under sum power constraint is derived using the circular channel approach. The optimal input power spectrum density shows an interesting geometrical waterfilling interpretation, where the frequency dependent power allocation of the relay nodes causes a wavy water level.