Consider the filter $y (n) + 0.25 y (n - 2) = 2 x (n) + 2 x (n - 1)$ . Then the steady -state value of the output y(n) due to the input $x (n) = 5 u (n)$ is ________
16

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Solution

The correct option is A 16
Apply Z.T. for the equation, we get

$Y (z) + 0.25 Y (z) z^{- 2} = 2 X (z) + 2 z^{- 1} X (z)$

$\frac{Y (z)}{X (z)} = H (z) = \frac{2 + 2 z^{- 1}}{1 + 0.25 z^{- 2}}$

$X (z) = \frac{5}{1 - z^{- 1}}$

$Y (z) = X (z) \cdot H (z) = \frac{2 + 2 z^{- 1}}{1 + 0.25 z^{- 2}} \times \frac{5}{1 - z^{- 1}}$

Applying final value theorem,

$y (\infty) = lim z \to 1 (1 - z^{- 1}) . Y (z)$

$= (\frac{2 + 2}{1 + 0.25}) \times 5 = \frac{4}{(5 / 4)} \times 5 = 16$

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