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3 (a) Express the signal x(t) as a sum of unit step functions for the time interval 3

3 (a) Express the signal x(t) as a sum of unit step functions for the time interval 3

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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Consider the following signal x(t) shown 

**Transcription for Educational Website** ### Problem Description #### Signal Representation The figure presents a piecewise function expressed in terms of unit step functions. The signal \( x(t) \) is defined over the time interval \( 3 < t < 6 \). 1. Between \( t = 0 \) and \( t = 3 \), the signal remains at 0. 2. At \( t = 3 \), the signal jumps to a value of 1. 3. Between \( t = 3 \) and \( t = 5 \), the signal remains at 1. 4. At \( t = 5 \), the signal drops to a value of -1. 5. Between \( t = 5 \) and \( t = 6 \), the signal remains at -1. 6. At \( t = 6 \), the signal returns to 0. #### Task (a) Express the signal \( x(t) \) as a sum of unit step functions for the time interval \( 3 < t < 6 \). #### Task (b) If \( x(t) \) is fed into a Linear Time-Invariant (LTI) system with a step response defined as: \[ g(t) = e^{-2t} u(t) \] determine the corresponding output without using convolution. ### Graph Explanation The graph illustrates the behavior of the signal \( x(t) \): - **Horizontal Axis (t):** Represents time in seconds (s). - **Vertical Axis:** Represents the amplitude of the signal, with values ranging from -1 to 1. - The step changes in the signal can be used to write the signal \( x(t) \) as a sum of shifted unit step functions. ### Approach to Solution - **Part (a):** Use the information from the graph to express \( x(t) \) as a series involving \( u(t) \), where \( u(t) \) represents the unit step function. The function \( x(t) \) can be expressed based on the changes at \( t = 3, 5, \) and \( 6 \). - **Part (b):** Employ properties of LTI systems and the step response to find the output, leveraging known transformations without explicitly computing a convolution.
Transcribed Image Text:**Transcription for Educational Website** ### Problem Description #### Signal Representation The figure presents a piecewise function expressed in terms of unit step functions. The signal \( x(t) \) is defined over the time interval \( 3 < t < 6 \). 1. Between \( t = 0 \) and \( t = 3 \), the signal remains at 0. 2. At \( t = 3 \), the signal jumps to a value of 1. 3. Between \( t = 3 \) and \( t = 5 \), the signal remains at 1. 4. At \( t = 5 \), the signal drops to a value of -1. 5. Between \( t = 5 \) and \( t = 6 \), the signal remains at -1. 6. At \( t = 6 \), the signal returns to 0. #### Task (a) Express the signal \( x(t) \) as a sum of unit step functions for the time interval \( 3 < t < 6 \). #### Task (b) If \( x(t) \) is fed into a Linear Time-Invariant (LTI) system with a step response defined as: \[ g(t) = e^{-2t} u(t) \] determine the corresponding output without using convolution. ### Graph Explanation The graph illustrates the behavior of the signal \( x(t) \): - **Horizontal Axis (t):** Represents time in seconds (s). - **Vertical Axis:** Represents the amplitude of the signal, with values ranging from -1 to 1. - The step changes in the signal can be used to write the signal \( x(t) \) as a sum of shifted unit step functions. ### Approach to Solution - **Part (a):** Use the information from the graph to express \( x(t) \) as a series involving \( u(t) \), where \( u(t) \) represents the unit step function. The function \( x(t) \) can be expressed based on the changes at \( t = 3, 5, \) and \( 6 \). - **Part (b):** Employ properties of LTI systems and the step response to find the output, leveraging known transformations without explicitly computing a convolution.
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