Math Problem Statement
Solution
To solve this problem using the graphical method, we need to perform the convolution of the input sequence with the impulse response of the LTI (Linear Time-Invariant) system. The output sequence is given by the convolution:
Steps for Graphical Convolution
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Identify Lengths of Sequences:
- The input sequence has a length of 2.
- The impulse response has a length of 3.
- The length of the output sequence will be .
-
Reverse the Sequence :
- Since we are using the graphical convolution method, we need to flip (even though for this finite sequence it doesn’t change much visually).
- Flipped is still .
-
Shift and Multiply:
- Shift for each time step , align it with , multiply element-wise, and sum the products for each shift.
Calculating for Each Shift
-
:
- Align the first element of with the first element of .
- Only the first element of and the first element of overlap.
- .
-
:
- Shift one position to the right.
- Now aligns with and aligns with .
- .
-
:
- Shift another position to the right.
- Now aligns with , and is out of range.
- .
-
:
- Shift one more position to the right.
- Now only overlaps with .
- .
Final Output Sequence
The output sequence is:
Would you like a more detailed explanation of each step, or do you have any questions?
Related Questions
- How does the graphical method differ from the analytical method of convolution?
- What is the significance of convolution in LTI systems?
- How would the output sequence change if the impulse response had a different length?
- Can we use the convolution property in the frequency domain instead of the time domain?
- How do properties like causality and stability affect the convolution process?
Tip
When using the graphical method, visualize each shift and overlap to simplify the calculations, especially for longer sequences.
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Math Problem Analysis
Mathematical Concepts
Linear Time-Invariant (LTI) Systems
Convolution
Graphical Convolution
Formulas
y(n) = x(n) * h(n)
Convolution sum y(n) = Σ x(k) * h(n-k)
Theorems
Convolution Theorem
Suitable Grade Level
Grades 11-12
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