We present a new proof of the characterization theorem for Minkowski Pythagorean-hodograph curves in the Minkowski spaces $mathbf{R}^{n+1,m}$. For an polynomial curves $mathbf{s}(t)=(x_1(t),...,;x_{n+m}(t))$, we also find Minkowski Pythagorean-hodograph curves $mathbf{r}(t)=(x_0(t),;x_1(t),...,;x_{n+m}(t))$. In case m=0, Minkowski Pythagorean-hodograph curves become Pythagorean-hodograph curves in the Euclidean spaces $mathbf{R}^{n+1}$ and Theorems in this paper hold for these Pythagorean-hodograph curves.