Exercise · ZW-DISCRETE-TRANSFER-FUNCTION-010 · ★★★ Advanced
A system satisfies \(y[n]=0.6\,y[n-1]-0.08\,y[n-2]+x[n]\) with zero initial conditions. What is \(H(z)\)?
Choose one option.
Show answer
- A. \(\frac{1}{1-0.6z^{-1}+0.08z^{-2}}\)
Explanation
Moving the recursive terms to the left gives \(Y(z)(1-0.6z^{-1}+0.08z^{-2})=X(z)\). The sign of the second-order term changes because of the original negative sign, which produces the option with the positive term in the denominator.
Review the concept: Discrete transfer function
The discrete transfer function \(H(z)\) is the Z-transform of the impulse response \(h[n]\) of a discrete-time LTI system, with \(H(z)=Y(z)/X(z)\) under zero initial conditions.
\[H(z)=\sum_{n=-\infty}^{\infty} h[n]\,z^{-n}\]
More practice problems: discrete transfer function
- ZW-DISCRETE-TRANSFER-FUNCTION-001 What does H(z) represent for a discrete LTI system?
- ZW-DISCRETE-TRANSFER-FUNCTION-002 If Y(z)=H(z)X(z), what time-domain operation corresponds to the product in z?
- ZW-DISCRETE-TRANSFER-FUNCTION-003 What is the name of the region in the z-plane where the transform defining H(z) converges?
- ZW-DISCRETE-TRANSFER-FUNCTION-004 For H(z)=(1)/(1-0.5z^-1), where is the pole?
- ZW-DISCRETE-TRANSFER-FUNCTION-005 What condition guarantees BIBO stability of a discrete system with a rational transfer function?
- ZW-DISCRETE-TRANSFER-FUNCTION-006 For y[n]=0.8 y[n-1]+x[n] with zero initial conditions, what is H(z)?
- ZW-DISCRETE-TRANSFER-FUNCTION-007 If h[n]=δ[n]+0.5 δ[n-1], what is H(z)?
- ZW-DISCRETE-TRANSFER-FUNCTION-008 A system has H(z)=(z)/(z-0.3). What is its causal impulse response h[n]?