Exercise · ZW-IMPULSE-RESPONSE-010 · ★★★ Advanced
Let \(h[n]=(0.5)^n u[n]\) and \(x[n]=u[n]\). Compute \(y[0]\) and \(y[1]\) by convolution and pick the correct option.
Choose one option.
Show answer
- D. y[0]=1, y[1]=1.5
Explanation
For \(n=1\), \(y[1]=\sum_k h[k]x[1-k]=h[0]+h[1]=1+0.5=1.5\), and for \(n=0\) we get \(y[0]=h[0]=1\). Forgetting the \(h[1]\) term in the sum leads to the option with 0.5 at \(y[1]\).
Review the concept: Impulse response
The impulse response \(h[n]\) is the output of an LTI system when its input is the unit impulse \(\delta[n]\).
\[y[n]=\sum_{k=-\infty}^{\infty} x[k]\,h[n-k]\]
More practice problems: impulse response
- ZW-IMPULSE-RESPONSE-001 What is the output of an LTI system called when the input is δ[n]?
- ZW-IMPULSE-RESPONSE-002 What value does δ[n] take at n=0?
- ZW-IMPULSE-RESPONSE-003 If h[n]=0 for n<0, what property does the system have?
- ZW-IMPULSE-RESPONSE-004 If h[n] is nonzero only for n=0,1,2, what is the system called?
- ZW-IMPULSE-RESPONSE-005 Which operation combines the input x[n] with the response h[n] to produce the output?
- ZW-IMPULSE-RESPONSE-006 A system has h[n]=(0.5)^n u[n]. What is h[2]?
- ZW-IMPULSE-RESPONSE-007 For h[n]=δ[n]+2δ[n-1] and input x[n]=δ[n], what is y[1]?
- ZW-IMPULSE-RESPONSE-008 A system has h[n]=u[n]-u[n-3]. What is h[1]?