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The input x(t) and system impulse response h(t) are given below. Determine the expression for the output y(t). x(t) y(t) LTI system h(t) x(t) 4h(t) 1 1 1 2

The input x(t) and system impulse response h(t) are given below. Determine the expression for the output y(t). x(t) y(t) LTI system h(t) x(t) 4h(t) 1 1 1 2

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Can you please solve this with graphical convolution/analytically instead of Laplace transforms for both parts. 

The input \( x(t) \) and system impulse response \( h(t) \) are given below. Determine the expression for the output \( y(t) \). **Diagram Description:** 1. **Block Diagram:** - There's a block labeled "LTI system \( h(t) \)" with an input arrow marked \( x(t) \) and an output arrow marked \( y(t) \). 2. **Input Signal \( x(t) \):** - The graph of \( x(t) \) is a rectangular pulse. - \( x(t) \) equals 1 for \( 0 \leq t \leq 1 \). - \( x(t) \) is 0 elsewhere. 3. **Impulse Response \( h(t) \):** - The graph of \( h(t) \) is an exponentially increasing function. - The function is \( e^t \) for \( 0 \leq t \leq 2 \). - \( h(t) \) is 0 for \( t < 0 \) and \( t > 2 \). **Problem:** - You need to calculate the output \( y(t) \) of the LTI system by convolving \( x(t) \) with \( h(t) \).
Transcribed Image Text:The input \( x(t) \) and system impulse response \( h(t) \) are given below. Determine the expression for the output \( y(t) \). **Diagram Description:** 1. **Block Diagram:** - There's a block labeled "LTI system \( h(t) \)" with an input arrow marked \( x(t) \) and an output arrow marked \( y(t) \). 2. **Input Signal \( x(t) \):** - The graph of \( x(t) \) is a rectangular pulse. - \( x(t) \) equals 1 for \( 0 \leq t \leq 1 \). - \( x(t) \) is 0 elsewhere. 3. **Impulse Response \( h(t) \):** - The graph of \( h(t) \) is an exponentially increasing function. - The function is \( e^t \) for \( 0 \leq t \leq 2 \). - \( h(t) \) is 0 for \( t < 0 \) and \( t > 2 \). **Problem:** - You need to calculate the output \( y(t) \) of the LTI system by convolving \( x(t) \) with \( h(t) \).
**Discrete Time System Analysis** Consider a discrete time system given by \[ y[n] + ay[n-1] = x[n] + bx[n-1] + x[n-2] \] If \( x[n] = \cos \left(\frac{2\pi}{3} n\right) + \cos \left(\frac{\pi}{2} n\right) \), determine the constants \( a \) and \( b \) so that the output is \[ y[n] = \frac{3}{4} \sin \left(\frac{\pi}{2} n + \theta \right) \] Additionally, what is the resulting phase \( \theta \)?
Transcribed Image Text:**Discrete Time System Analysis** Consider a discrete time system given by \[ y[n] + ay[n-1] = x[n] + bx[n-1] + x[n-2] \] If \( x[n] = \cos \left(\frac{2\pi}{3} n\right) + \cos \left(\frac{\pi}{2} n\right) \), determine the constants \( a \) and \( b \) so that the output is \[ y[n] = \frac{3}{4} \sin \left(\frac{\pi}{2} n + \theta \right) \] Additionally, what is the resulting phase \( \theta \)?
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