Exercise · ZW-DISCRETE-TRANSFER-FUNCTION-007 · ★★☆ Intermediate
If \(h[n]=\delta[n]+0.5\,\delta[n-1]\), what is \(H(z)\)?
Choose one option.
Show answer
- D. \(1+0.5z^{-1}\)
Explanation
The transform of \(\delta[n-k]\) is \(z^{-k}\), so each term contributes its negative power of \(z\). The option with a positive power uses a shift in the opposite direction.
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-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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