Exercise · ZW-LINEAR-TIME-INVARIANT-SYSTEMS-010 · ★★★ Advanced
An LTI system has \(h[n]=\left(\tfrac{1}{2}\right)^n u[n]\). For the input \(x[n]=\delta[n]+\delta[n-1]\), what is \(y[1]\)?
Choose one option.
Show answer
- A. \(\tfrac{3}{2}\)
Explanation
The output is \(y[n]=h[n]+h[n-1]\), so \(y[1]=h[1]+h[0]=\tfrac{1}{2}+1=\tfrac{3}{2}\). Only \(\tfrac{3}{2}\) matches this sum of two terms.
Review the concept: Linear time-invariant systems
A linear time-invariant (LTI) system is a system that satisfies superposition and whose behavior does not change when the input is shifted. It is determined by its impulse response \(h[n]\).
\[y[n]=\sum_{k=-\infty}^{\infty} x[k]\,h[n-k]\]
More practice problems: linear time-invariant systems
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-001 Which of these properties is required for a system to be LTI?
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-002 A system is time-invariant if a shift of the input produces:
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-003 What is the impulse response of an identity system, where y[n]=x[n]?
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-004 A system with y[n]=x[n]^2 is linear.
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-005 Which operation is used to compute the output of an LTI system from its input and its impulse respo…
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-006 If h[n]=δ[n]+δ[n-2] and x[n]=δ[n-1], what is y[n]?
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-007 If h[n]=(tfrac12)^n u[n] and x[n]=δ[n], what is y[2]?
- ZW-LINEAR-TIME-INVARIANT-SYSTEMS-008 If x[n]=u[n] and h[n]=δ[n]-δ[n-1], what is y[n]?