= Solution
The <seasonal difference operator> has $b_0=1$, $b_{12}=-1$ and all other coefficients zero. Its frequency response is $1-e^{12i\lambda}$, so
$$
\boxed{f_Z(\lambda)=4\sin^2(6\lambda)f_Y(\lambda),\qquad
G_b(\lambda)=2|\sin(6\lambda)|.}
$$
The right sketch shows zeros at $\lambda=j\pi/6$ for $j=0,\ldots,6$, and maxima of height two at $(2j+1)\pi/12$ for $j=0,\ldots,5$. The <seasonal difference operator> removes every exactly twelve-periodic component, including a constant. Its <filter gain> has a comb pattern: it selectively suppresses seasonal frequencies, rather than increasing monotonically with frequency.
Back to article page
