Joint SVD-GSVD precoding technique and secrecy capacity lower bound for the MIMO relay wire-tap channel

We consider a problem of secure communications for the communication system consisting of multiple outputs for a source and a relay and multiple inputs for the relay, a destination and an eavesdropper. For the above-mentioned communication system, we establish a lower bound on the secrecy capacity at which secure communications between the source and the destination are attainable. We make use of the singular value decomposition (SVD) and its generalization to decompose the whole system into parallel independent channels. At the source, the generalized singular value decomposition (GSVD) is performed to simultaneously diagonalize the channel matrices of the relay and the destination and independently code across the resulting parallel channels. At the relay, the SVD is performed to beamform the signal towards the destination. The scalar case of what we are considering in this paper has been investigated in previous literature, to prove that the introduction of a fourth party, the relay, in the wire-tap channel facilitates secure wireless communications. Our simulation results are in line with the scalar case's and prove to be successful in achieving secrecy capacity where the conventional model failed, i.e. when no relay is introduced and the eavesdropper's channel incurs as little noise as the legitimate receiver.