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hi[n] h2[n] h3[n] DT LTI system with subsystems' impulse responses h, [n] = 26[n], hz[n] = 0.2"u[n – 2], hz[n] = (-0.5)"u[n] What is the step response?

hi[n] h2[n] h3[n] DT LTI system with subsystems' impulse responses h, [n] = 26[n], hz[n] = 0.2"u[n – 2], hz[n] = (-0.5)"u[n] What is the step response?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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On this educational webpage, we will examine a Discrete-Time (DT) Linear Time-Invariant (LTI) system with three subsystems, each governed by a specific impulse response. The system is illustrated by a block diagram: 1. An input signal enters a summing junction. 2. The output of the summing junction splits into two paths: - One path connects to the subsystem with impulse response \( h_1[n] \). - The other path leads directly down to the subsystem with impulse response \( h_3[n] \). 3. The output of the subsystem \( h_1[n] \) leads to another subsystem with impulse response \( h_2[n] \). 4. The output from subsystem \( h_2[n] \) serves as the final output of the system. 5. Meanwhile, the output from subsystem \( h_3[n] \) loops back as a feedback to the initial summing junction. The impulse responses of the subsystems are defined as follows: \[ h_1[n] = 2\delta[n], \] \[ h_2[n] = 0.2^n u[n-2], \] \[ h_3[n] = (-0.5)^n u[n], \] where: - \( \delta[n] \) is the discrete-time delta function, - \( u[n] \) is the unit step function, - \( u[n-2] \) is the unit step function delayed by 2 units. ### Problem: Determine the step response of this LTI system. ### Solution Steps: You would typically follow these steps to determine the system's step response: 1. Find the overall system impulse response by analyzing how the impulse responses \( h_1[n] \), \( h_2[n] \), and \( h_3[n] \) interact. 2. Convolve the system's overall impulse response with the unit step function \( u[n] \). ### Detailed Explanation of Impulse Responses: - **Impulse Response 1:** \( h_1[n] = 2\delta[n] \) This indicates that subsystem 1 scales the input by 2 without any delay. - **Impulse Response 2:** \( h_2[n] = 0.2^n u[n-2] \) This shows that the response starts at \( n = 2 \) and decays exponentially with a factor of
Transcribed Image Text:On this educational webpage, we will examine a Discrete-Time (DT) Linear Time-Invariant (LTI) system with three subsystems, each governed by a specific impulse response. The system is illustrated by a block diagram: 1. An input signal enters a summing junction. 2. The output of the summing junction splits into two paths: - One path connects to the subsystem with impulse response \( h_1[n] \). - The other path leads directly down to the subsystem with impulse response \( h_3[n] \). 3. The output of the subsystem \( h_1[n] \) leads to another subsystem with impulse response \( h_2[n] \). 4. The output from subsystem \( h_2[n] \) serves as the final output of the system. 5. Meanwhile, the output from subsystem \( h_3[n] \) loops back as a feedback to the initial summing junction. The impulse responses of the subsystems are defined as follows: \[ h_1[n] = 2\delta[n], \] \[ h_2[n] = 0.2^n u[n-2], \] \[ h_3[n] = (-0.5)^n u[n], \] where: - \( \delta[n] \) is the discrete-time delta function, - \( u[n] \) is the unit step function, - \( u[n-2] \) is the unit step function delayed by 2 units. ### Problem: Determine the step response of this LTI system. ### Solution Steps: You would typically follow these steps to determine the system's step response: 1. Find the overall system impulse response by analyzing how the impulse responses \( h_1[n] \), \( h_2[n] \), and \( h_3[n] \) interact. 2. Convolve the system's overall impulse response with the unit step function \( u[n] \). ### Detailed Explanation of Impulse Responses: - **Impulse Response 1:** \( h_1[n] = 2\delta[n] \) This indicates that subsystem 1 scales the input by 2 without any delay. - **Impulse Response 2:** \( h_2[n] = 0.2^n u[n-2] \) This shows that the response starts at \( n = 2 \) and decays exponentially with a factor of
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