  In this work we study the secrecy capacity of Gaussian multiple-input
multiple-output (MIMO) wiretap channels (WTCs) with a finite memory, subject to
a per-symbol average power constraint on the MIMO channel input. MIMO channels
with finite memory are very common in wireless communications as well as in
wireline communications (e.g., in communications over power lines). To derive
the secrecy capacity of the Gaussian MIMO WTC with finite memory we first
construct an asymptotically-equivalent block-memoryless MIMO WTC, which is then
transformed into a set of parallel, independent, memoryless MIMO WTCs in the
frequency domain. The secrecy capacity of the Gaussian MIMO WTC with finite
memory is obtained as the secrecy capacity of the set of parallel independent
memoryless MIMO WTCs, and is expressed as a maximization over the input
covariance matrices in the frequency domain. Lastly, we detail two applications
of our result: First, we show that the secrecy capacity of the Gaussian scalar
WTC with finite memory can be achieved by waterfilling, and obtain a
closed-form expression for this secrecy capacity. Then, we use our result to
characterize the secrecy capacity of narrowband powerline channels, thereby
resolving one of the major open issues for this channel model.
