On the Distortion of the Eigenvalue Spectrum in MIMO Amplify-and-Forward Multi-Hop Channels

Consider a wireless MIMO multi-hop channel with ns non-cooperating source antennas and nd fully cooperating destination antennas, as well as L clusters containing k non-cooperating relay antennas each. The source signal traverses all L clusters of relay antennas, before it reaches the destination. When relay antennas within the same cluster scale their received signals by the same constant before the retransmission, the equivalent channel matrix H relating the input signals at the source antennas to the output signals at the destination antennas is proportional to the product of channel matrices Hl, l = 1, . . . , L + 1, corresponding to the individual hops. We perform an asymptotic capacity analysis for this channel as follows: In a first instance we take the limits ns → ∞, nd → ∞ and k → ∞, but keep both ns/nd and k/nd fixed. Then, we take the limits L → ∞ and k/nd → ∞. Requiring that the Hl’s satisfy the conditions needed for the Marčenko-Pastur law, we prove that the capacity scales linearly in min{ns, nd}, as long as the ratio k/nd scales at least linearly in L. Moreover, we show that up to a noise penalty and a pre-log factor the capacity of a point-to-point MIMO channel is approached, when this scaling is slightly faster than linear. Conversely, almost all spatial degrees of freedom vanish for less than linear scaling.