Question

Determine the transfer function, $H(z) = \frac{Y(z)}{X(z)}$, of the discrete-time LTI system y[n+1] + y[n] - 0.5y[n - 2] = x[n] + 0.25x[n - 1]. $H(z) = \frac{z^{-1} - 0.5z^2}{1 - 0.25z^{-1}}$ $H(z) = \frac{1 + z - 0.5z^3}{z + 0.25z^2}$ $H(z) = \frac{z^3 + z^2 - 0.5}{z^2 + 0.25z}$ $H(z) = \frac{z + 0.25z^2}{1 + z - 0.5z^3}$ $H(z) = \frac{z^2 + 0.25z}{z^3 + z^2 - 0.5}$ $H(z) = \frac{1 - 0.25z^{-1}}{z^{-1} - 0.5z^2}$

          Determine the transfer function, $H(z) = \frac{Y(z)}{X(z)}$, of the discrete-time LTI system
y[n+1] + y[n] - 0.5y[n - 2] = x[n] + 0.25x[n - 1].

$H(z) = \frac{z^{-1} - 0.5z^2}{1 - 0.25z^{-1}}$

$H(z) = \frac{1 + z - 0.5z^3}{z + 0.25z^2}$

$H(z) = \frac{z^3 + z^2 - 0.5}{z^2 + 0.25z}$

$H(z) = \frac{z + 0.25z^2}{1 + z - 0.5z^3}$

$H(z) = \frac{z^2 + 0.25z}{z^3 + z^2 - 0.5}$

$H(z) = \frac{1 - 0.25z^{-1}}{z^{-1} - 0.5z^2}$

Determine the transfer function, H(z) = (Y(z))/(X(z)), of the discrete-time LTI system
y[n+1] + y[n] - 0.5y[n - 2] = x[n] + 0.25x[n - 1].

H(z) = (z^-1 - 0.5z^2)/(1 - 0.25z^-1)

H(z) = (1 + z - 0.5z^3)/(z + 0.25z^2)

H(z) = (z^3 + z^2 - 0.5)/(z^2 + 0.25z)

H(z) = (z + 0.25z^2)/(1 + z - 0.5z^3)

H(z) = (z^2 + 0.25z)/(z^3 + z^2 - 0.5)

H(z) = (1 - 0.25z^-1)/(z^-1 - 0.5z^2)

Added by Patrick F.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Aakash Goyal

Step 1

5y(z)z^-2 = x(z) + 0.25x(z)z^-1 Now, let's rearrange the equation to solve for y(z): y(z)(z + 1 - 0.5z^-2) = x(z) + 0.25x(z)z^-1 y(z) = (x(z) + 0.25x(z)z^-1) / (z + 1 - 0.5z^-2) Now, let's simplify the expression: y(z) = x(z)(1 + 0.25z^-1) / (z + 1 - 0.5z^-2) Show more…

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Determine the transfer function H(z) of the discrete-time LTI system X(z) y[n+1]+y[n]-0.5y[n-2]=x[n]+0.25x[n-1] H(z)=-1-0.5z+1-0.25z^-1 H(z)=1+2z-0.5z^2 +0.25z H(z)= +0.25z^2 +z-0.53 OH(z)= +0.25z^3+2z-0.5 =H0 1-0.25z^-1 z^-1-0.52