  This paper studies a full-duplex filter-and-forward (FD-FF) relay system in
frequency-selective channels. Conventionally, the loop-back signal at the FD
relay is treated as harmful self-interference and needs to be significantly
suppressed via both analog- and digital-domain cancellation. However, the
performance of the conventional self-interference cancellation approach is
fundamentally limited due to the quantization error induced by the
analog-to-digital converter (ADC) with limited dynamic range. In this paper, we
consider an analog filter-and-forward design to help avoid the quantization
error, and surprisingly show that the maximum achievable rate of such an FD-FF
relay system is in fact regardless of the loop-back channel at the FD relay. We
characterize the maximum achievable rate of this channel by jointly optimizing
the transmit power allocation over frequency at the source and the frequency
response of the filter at the relay, subject to their individual power
constraints. Although this problem is non-convex, we obtain its optimal
solution by applying the Lagrange duality method. By simulations it is shown
that the proposed joint source and relay optimization achieves rate gains over
other heuristic designs, and is also advantageous over the conventional
approach by cancelling the relay loop-back signal as self-interference,
especially when the residual self-interference after cancellation is still
significant.
