Problem $1$: Consider the implementation of a $\backslash (2^\{\backslash$text $\{$nd $\}\}\backslash )$ order filter shown in Figure $1.$ Scale the filter structure such that no intermediate signal node overloads and the overall transfer function remains the same for each of the following cases.

$($a$)$ Input is constrained to be a sinusoid of arbitrary frequency and amplitude that can be less than or equal to unity.

$($b$)$ Input is a white noise sequence with arbitrary variance bounded to less than or equal to unity.

$($c$)$ Which of the above scenarios will result in a lower round$-$off

Figure $1$: A $\backslash (\{\}^\{\backslash$text $\{$nd $\}\}\backslash )$ order digital filter implementation.

noise at the output? Explain quantitatively.

Notes:

$(1)$ The pass$-$band gain of the filter is more than $1.$ You can assume that the output has as many additional MSB $($integer$)$ bits as needed to accommodate the output range, but the same number of fractional bits as all other signals.

$\backslash [$

$(5+5+5=15\backslash$text $\{$ points $\})$

$\backslash ]$