A polyphase filter used in a decimation$-$by$-$four process is depicted with the structure shown here. The $H_k(z)$ blocks represent tapped$-$delay line FIR polyphase subfilters containing unit$-$delay elements, multipliers, and adders.

$($a$)$ Why are polyphase subfilters useful when used in a decimation process? $(5$ points$)$

$($b$)$ Determine how to replace the commutating $($rotating$)$ input switch in the figure above using only the delay and downsampler elements shown below on the left. That is$,$ determine what interconnection of delay and downsampler elements must be inside the mysterious block below on the right to make that figure equivalent to the original figure above. $(10$ points$)$

Hint: Given some $x(n)$ sequence, write the $x(n),x(n-1),x(n-2),$ etc., sample sequences on the four lines driving the $H_k(z)$ subfilters in the first figure above. Then determine how to obtain those same sample sequences for routing to the subfilters in the figure directly above on the right.