  If a fraction $f_{\rm dcdm}$ of the Dark Matter decays into invisible and
massless particles (so-called "dark radiation") with the decay rate (or inverse
lifetime) $\Gamma_{\rm dcdm}$, such decay will leave distinctive imprints on
cosmological observables. With a full consideration of the Boltzmann hierarchy,
we calculate the decay-induced impacts not only on the CMB but also on the
redshift distortion and the kinetic Sunyaev-Zel'dovich effect, while providing
detailed physical interpretations based on evaluating the evolution of
gravitational potential. By using the current cosmological data with a
combination of Planck 2015, Baryon Acoustic Oscillation and redshift distortion
measurements which can improve the constraints, we update the $1\sigma$ bound
on the fraction of decaying DM from $f_{\rm dcdm}\lesssim5.26\%$ to $f_{\rm
dcdm}\lesssim2.73\%$ for the short-lived DM (assuming $\Gamma_{\rm
dcdm}/H_0\gtrsim10^4$). However, no constraints are improved from RSD data
($f_{\rm dcdm}\lesssim0.94\%$) for the long-lived DM (i.e., $\Gamma_{\rm
dcdm}/H_0\lesssim10^4$). We also find the fractional DM decay can only slightly
reduce the $H_0$ and $\sigma_8$ tensions, which is consistent with other
previous works. Furthermore, our calculations show that the kSZ effect in
future would provide a further constraining power on the decaying DM.
