  Shannon-Hartley theorem can accurately calculate the channel capacity when
the signal observation time is infinite. However, the calculation of
finite-time mutual information, which remains unknown, is essential for guiding
the design of practical communication systems. In this paper, we investigate
the mutual information between two correlated Gaussian processes within a
finite-time observation window. We first derive the finite-time mutual
information by providing a limit expression. Then we numerically compute the
mutual information within a single finite-time window. We reveal that the
number of bits transmitted per second within the finite-time window can exceed
the mutual information averaged over the entire time axis, which is called the
exceed-average phenomenon. Furthermore, we derive a finite-time mutual
information formula under a typical signal autocorrelation case by utilizing
the Mercer expansion of trace class operators, and reveal the connection
between the finite-time mutual information problem and the operator theory.
Finally, we analytically prove the existence of the exceed-average phenomenon
in this typical case, and demonstrate its compatibility with the Shannon
capacity.
