  This paper discusses model order reduction of LTI systems over limited
frequency intervals within the framework of balanced truncation. Two new
\emph{frequency-dependent balanced truncation} methods were developed, one is
\emph{SF-type frequency-dependent balanced truncation} to copy with the cases
that only a single dominating point of the operating frequency interval is
pre-known, the other is \emph{interval-type frequency-dependent balanced
truncation} to deal with the cases that both of the upper and lower bound of
frequency interval are known \emph{a priori}. SF-type error bound and
interval-type error bound are derived for the first time to estimate the
desired approximation error over pre-specified frequency interval. We show that
the new methods generally lead to good in-band approximation performance, at
the same time, provide accurate error bounds under certain conditions. Examples
are included for illustration.
