Exercise · ZW-DISCRETE-TRANSFER-FUNCTION-008 · ★★☆ Intermediate
A system has \(H(z)=\frac{z}{z-0.3}\). What is its causal impulse response \(h[n]\)?
Choose one option.
Show answer
- C. \(0.3^n u[n]\)
Explanation
Writing \(H(z)=\frac{1}{1-0.3z^{-1}}\) reveals a causal geometric series with ratio 0.3. The option with inverted powers corresponds to a pole outside the unit circle, with a different region of convergence.
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-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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