Exercise · ZW-DISCRETE-TRANSFER-FUNCTION-006 · ★★☆ Intermediate
For \(y[n]=0.8\,y[n-1]+x[n]\) with zero initial conditions, what is \(H(z)\)?
Choose one option.
Show answer
- A. \(\frac{1}{1-0.8z^{-1}}\)
Explanation
Transforming gives \(Y(z)(1-0.8z^{-1})=X(z)\), so \(H(z)=1/(1-0.8z^{-1})\). The option with a positive sign in the denominator drops the sign change when moving the recursive term.
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-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]?
- ZW-DISCRETE-TRANSFER-FUNCTION-009 For H(z)=(1)/(1-1.2z^-1) with region of convergence |z|>1.2, is the causal system stable?
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