= Solution
The <composition of absolutely summable time-series filters> is $(1-B^{12})(1-B)$, with $B$ the <backshift operator>. Thus
$$
\boxed{Z_t=X_t-X_{t-1}-X_{t-12}+X_{t-13},}
$$
and its only nonzero coefficients are $c_0=1$, $c_1=-1$, $c_{12}=-1$, $c_{13}=1$. Multiplication of the two <filter gains> gives
$$
\boxed{G_c(\lambda)=4\sin(\lambda/2)|\sin(6\lambda)|,\quad0\leq\lambda\leq\pi.}
$$
Equivalently, its <spectral density transformation under a linear filter> is $f_Z=16\sin^2(\lambda/2)\sin^2(6\lambda)f_X$.
Back to article page
