  The relaxation function theory of doped two-dimensional $S=1/2$ Heisenberg
antiferromagnetic (AF) system in the paramagnetic state is presented taking
into account the hole subsystem as well as both the electron and AF
correlations. The expression for fourth frequency moment of relaxation shape
function is derived within the $t-J$ model. The presentation obeys rotational
symmetry of the spin correlation functions and is valid for all wave vectors
through the Brillouin zone. The spin diffusion contribution to relaxation rates
is evaluated and is shown to play a significant role in carrier free and doped
antiferromagnet in agreement with exact diagonalization calculations. At low
temperatures the main contribution to the nuclear spin-lattice relaxation rate,
$^{63}(1/T_1)$, of plane $^{63}$Cu arises from the AF fluctuations, and
$^{17}(1/T_1)$, of plane $^{17}$O, has the contributions from the wavevectors
in the vicinity of $(\pi,\pi)$ and small $q \sim 0$. It is shown that the
theory is able to explain the main features of experimental data on temperature
and doping dependence of $^{63}(1/T_1)$ in the paramagnetic state of both
carrier free La$_2$CuO$_4$ and doped La$_{2-x}$Sr$_x$CuO$_4$ compounds.
