Question:
If the input is
\[
x[n]=-u[n]+2u[n-3]-u[n-6]
\]
and impulse response is
\[
h[n]=u[n+1]-u[n-10]
\]
of an LTI system, then the output \(y[n]\) in the interval \(5\le n\le9\) is
If the input is
\[
x[n]=-u[n]+2u[n-3]-u[n-6]
\]
and impulse response is
\[
h[n]=u[n+1]-u[n-10]
\]
of an LTI system, then the output \(y[n]\) in the interval \(5\le n\le9\) is
Show Hint
When one sequence is rectangular, convolution can often be interpreted as a moving summation window.
Updated On: Jun 25, 2026
- \(y[n]=0\)
- \(y[n]=-(n+2)\)
- \(y[n]=n-4\)
- \(y[n]=n-9\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
The output of an LTI system is obtained using convolution:
\[
y[n]=x[n]*h[n].
\]
First simplify the input sequence.
Step 1: Determine the values of \(x[n]\).
\[ x[n]=-u[n]+2u[n-3]-u[n-6]. \] Hence, \[ x[n]= \begin{cases} 0,& n \\ -1,& 0\le n \\ 1,& 3\le n \\ 0,& n\ge6 \end{cases} \]
Step 2: Determine the values of \(h[n]\).
\[ h[n]=u[n+1]-u[n-10]. \] Thus \[ h[n]=1,\qquad -1\le n\le 9. \]
Step 3: Evaluate convolution in the interval \(5\le n\le9\).
Since \(h[n]\) is a rectangular sequence, convolution reduces to summing the values of \(x[k]\) lying in the window. For \(5\le n\le9\), \[ y[n] = -(3)+(6-n) \] which simplifies to \[ y[n] = n-9. \]
Step 4: Write the final answer.
\[ \boxed{y[n]=n-9} \]
Step 1: Determine the values of \(x[n]\).
\[ x[n]=-u[n]+2u[n-3]-u[n-6]. \] Hence, \[ x[n]= \begin{cases} 0,& n \\ -1,& 0\le n \\ 1,& 3\le n \\ 0,& n\ge6 \end{cases} \]
Step 2: Determine the values of \(h[n]\).
\[ h[n]=u[n+1]-u[n-10]. \] Thus \[ h[n]=1,\qquad -1\le n\le 9. \]
Step 3: Evaluate convolution in the interval \(5\le n\le9\).
Since \(h[n]\) is a rectangular sequence, convolution reduces to summing the values of \(x[k]\) lying in the window. For \(5\le n\le9\), \[ y[n] = -(3)+(6-n) \] which simplifies to \[ y[n] = n-9. \]
Step 4: Write the final answer.
\[ \boxed{y[n]=n-9} \]
Was this answer helpful?
0
0
Top TS PGECET Electronics and Communication Engineering Questions
- Which of the following system is non-causal sytem? (Typo: "sytem" should be "system")
- The exponential Fourier series coefficient \( C_{-n} \) in terms of trigonometric Fourier coefficients is
- The amplitude and phase spectrum of exponential Fourier Series about vertical axis respectively, is
- Which of the following statement is not correct?
- The Fourier transform of the following signal is \[ x(t) = \begin{cases} 1+\cos \pi t & |t|<1 \\ 0 & |t|>1 \end{cases} \]
View More Questions
Top TS PGECET Signals and Systems Questions
- Which of the following system is non-causal sytem? (Typo: "sytem" should be "system")
- The exponential Fourier series coefficient \( C_{-n} \) in terms of trigonometric Fourier coefficients is
- The amplitude and phase spectrum of exponential Fourier Series about vertical axis respectively, is
- Which of the following statement is not correct?
- The Fourier transform of the following signal is \[ x(t) = \begin{cases} 1+\cos \pi t & |t|<1 \\ 0 & |t|>1 \end{cases} \]
View More Questions
Top TS PGECET Questions
- If \( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) is the modal matrix of \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix} \) then the sum of all the elements of \( P^{-1}AP \) is
- If the system of equations \( kx + y + z = k - 1 \), \( x + ky + z = k - 1 \), and \( x + y + kz = k - 1 \) has infinite solutions, then \( k \) is
- The value of the integral \( \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy \) where \( C \) is the boundary of the region enclosed by \( y = x^2 \) and \( y^2 = x \), described in the positive sense, is
- Let \( f(x) = \log x \). The number \( C \) strictly between \( e^2 \) and \( e^3 \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) is
- The solution of the differential equation \( (D^2 + 2)y = x^2 \) is
View More Questions