Consider the LTI system with the input-output relationship n-4 y[n] = x[n] + 3x[n – 2] + > sin((k – n)n/4) · x[k] k=-0 If h[:] is the system's impulse response, determine the following values: h[2], h[3], and h[7].
Consider the LTI system with the input-output relationship n-4 y[n] = x[n] + 3x[n – 2] + > sin((k – n)n/4) · x[k] k=-0 If h[:] is the system's impulse response, determine the following values: h[2], h[3], and h[7].
Introductory Circuit Analysis (13th Edition)
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Publisher:Robert L. Boylestad
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![**LTI System Analysis**
Consider the LTI system with the input-output relationship given by:
\[ y[n] = x[n] + 3x[n - 2] + \sum_{k=-\infty}^{n-4} \sin((k - n) \pi / 4) \cdot x[k] \]
If \( h[\cdot] \) is the system’s impulse response, determine the following values: \( h[2] \), \( h[3] \), and \( h[7] \).
---
### Explanation:
Here, \( y[n] \) is the output of the system and \( x[n] \) is the input to the system. The expression shows a linear combination of the input at different time indices, modified by constants and a summation term involving a sinusoidal function.
To determine the impulse response values, you would apply the definition of impulse response \( h[n] \) in the context of LTI systems, which typically involves analyzing the system's response to a discrete delta impulse input \( \delta[n] \).
### Steps to Determine Impulse Response:
1. Substitute \( x[n] = \delta[n] \) into the input-output relationship.
2. Compute \( y[n] \) for the given cases \( [2] \), \( [3] \), and \( [7] \).
Solving this will yield the desired \( h[2] \), \( h[3] \), and \( h[7] \) values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd07e8233-789e-4760-ba71-e70869af7b54%2F6687c65c-0a15-402e-8c08-54d14972b9d4%2Fzbuawwm_processed.png&w=3840&q=75)
Transcribed Image Text:**LTI System Analysis**
Consider the LTI system with the input-output relationship given by:
\[ y[n] = x[n] + 3x[n - 2] + \sum_{k=-\infty}^{n-4} \sin((k - n) \pi / 4) \cdot x[k] \]
If \( h[\cdot] \) is the system’s impulse response, determine the following values: \( h[2] \), \( h[3] \), and \( h[7] \).
---
### Explanation:
Here, \( y[n] \) is the output of the system and \( x[n] \) is the input to the system. The expression shows a linear combination of the input at different time indices, modified by constants and a summation term involving a sinusoidal function.
To determine the impulse response values, you would apply the definition of impulse response \( h[n] \) in the context of LTI systems, which typically involves analyzing the system's response to a discrete delta impulse input \( \delta[n] \).
### Steps to Determine Impulse Response:
1. Substitute \( x[n] = \delta[n] \) into the input-output relationship.
2. Compute \( y[n] \) for the given cases \( [2] \), \( [3] \), and \( [7] \).
Solving this will yield the desired \( h[2] \), \( h[3] \), and \( h[7] \) values.
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