  We present here new medium resolution spectra ($\lambda/\Delta\lambda$ $\sim$
1200) of K$-$M giants covering wavelength range 1.50$-$1.80 and 1.95$-$2.45
$\mu$m. The sample includes 72 K0$-$M8 giants from our TIRSPEC observations and
all available 35 giants in that spectral range from archival IRTF spectral
library. We have calibrated here the empirical relations between fundamental
parameters (e.g., effective temperature, surface gravity) and equivalent widths
of some important spectral features like Si I, Na I, Ca I, $^{12}$CO. We find
that the $^{12}$CO first overtone band at 2.29 $\mu$m and second overtone band
at 1.62 $\mu$m are a reasonably good indicator of temperature above 3400 K and
surface gravity. We show that the dispersion of empirical relations between
$^{12}$CO and $T_{eff}$ significantly improve considering the effect of surface
gravity.
