Question

The impulse response h(n) of the causal LTI discreate time system. That satisfying the difference equation $y\left(n\right)-0.25y(n-3)=4x\left(n\right)-x(n-3)$ is equal to P$\delta \left(n\right)$. Then the value of P is

• A. $\frac{1}{4}$
• B. 2
• C. $\frac{1}{2}$
• D. 4

Answer 1

The correct option is D 4

$y\left(n\right)-0.25y(n-3)=4x\left(n\right)-x(n-3)$

Applying Z-transform, we get

$(1-\frac{1}{4}{z}^{-3})Y\left(z\right)=(4-z^{-3})X\left(z\right)$

$H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}=\frac{(4-z^{-3})}{(1-\frac{1}{4}{z}^{-3})}=4$

$\therefore$ applying inverse Z- transform, we get,

$h\left(n\right)=4\delta \left(n\right)$

$\therefore P=4$