  We present a high precision frequency determination method for digitized NMR
FID signals. The method employs high precision numerical integration rather
than simple summation as in many other techniques. With no independent
knowledge of the other parameters of a NMR FID signal (phase $\phi$, amplitude
$A$, and transverse relaxation time $T_{2}$) this method can determine the
signal frequency $f_{0}$ with a precision of $1/(8\pi^{2}f_{0}^{2}T_{2}^{2})$
if the observation time $T$ is long enough. The method is especially convenient
when the detailed shape of the observed FT NMR spectrum is not well defined.
When $T_{2}$ is $+\infty$ and the signal becomes pure sinusoidal, the precision
of the method is $3/(2\pi^{2}f_{0}^{2}T^{2})$ which is one order more precise
than a typical frequency counter. Analysis of this method shows that the
integration reduces the noise by bandwidth narrowing as in a lock-in amplifier,
and no extra signal filters are needed. For a pure sinusoidal signal we find
from numerical simulations that the noise-induced error in this method reaches
the Cramer-Rao Lower Band(CRLB) on frequency determination. For the damped
sinusoidal case of most interest, the noise-induced error is found to be within
a factor of 2 of CRLB when the measurement time $T$ is a few times larger than
$T_{2}$.We discuss possible improvements for the precision of this method.
