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We study K-user M-hop Gaussian relay networks with Km nodes in the m-th layer, where M is even and K = K1 = KM+1. We observe that the time-varying nature of wireless channels (fading) can be exploited to mitigate the inter-user interference. The proposed block Markov encoding and relaying scheme exploits such channel variations and works for any isotropically distributed channels including Rayleigh fading. We show a general achievable degrees of freedom (DoF) region of this class of Gaussian relay networks, which coincides with the cut-set outer bound if M/Kmin is an integer, where Kmin = minm {Km}. Therefore, we completely characterize the DoF region for the case where M/Kmin is an integer.

Approximate capacity of a class of multi-source Gaussian relay networks