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The frequency response of an LTI system is shown below. For an input signal x(t) = cos(4t), what is output y(t)? -2 H(jw) 21 2 W Oy(t) = 0 Oy(t) = cos(4t)/2 Oy(t) = cos(4t) Oy(t) = 2cos(41) 27

The frequency response of an LTI system is shown below. For an input signal x(t) = cos(4t), what is output y(t)? -2 H(jw) 21 2 W Oy(t) = 0 Oy(t) = cos(4t)/2 Oy(t) = cos(4t) Oy(t) = 2cos(41) 27

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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LTI SYSTEM FREQUENCY RESPONSE ( NEED NEAT HANDWRITTEN SOLUTION ONLY OTHERWISE DOWNVOTE).

**Analyzing the Frequency Response of an LTI System** The frequency response of a Linear Time-Invariant (LTI) system is illustrated in the graph below. The task is to determine the output \(y(t)\) for a given input signal \(x(t) = \cos(4t)\). **Graph Explanation:** - The graph displays \(H(j\omega)\), the frequency response of the LTI system. - The x-axis represents the angular frequency \(\omega\), ranging from \(-2\) to \(2\). - The y-axis shows the magnitude of \(H(j\omega)\). - The graph is a rectangular function with: - A height (magnitude) of 2. - Nonzero values only within the range \(-2 \leq \omega \leq 2\). **Given Input Signal:** \[x(t) = \cos(4t)\] **Objective:** Determine the output signal \(y(t)\) of the LTI system. **Answer Choices:** 1. \( y(t) = 0 \) 2. \( y(t) = \cos(4t)/2 \) 3. \( y(t) = \cos(4t) \) 4. \( y(t) = 2\cos(4t) \) **Understanding the Frequency Response:** The input signal \(x(t) = \cos(4t)\) can be expressed in the frequency domain, involving frequencies \(\pm 4\). Evaluating \(H(j\omega)\) at these frequencies (\(\omega = 4\) and \(\omega = -4\)): - From the graph, \(H(j\omega)\) is zero outside the range \(-2 \leq \omega \leq 2\). - Therefore, the input frequencies \(\pm 4\) are outside the effective range of \(H(j\omega)\). - The system will attenuate these frequencies to zero. **Conclusion:** Given the frequency response and the provided input signal, the system output \(y(t) = 0\). Thus, the correct answer is: \[ \boxed{y(t) = 0} \]
Transcribed Image Text:**Analyzing the Frequency Response of an LTI System** The frequency response of a Linear Time-Invariant (LTI) system is illustrated in the graph below. The task is to determine the output \(y(t)\) for a given input signal \(x(t) = \cos(4t)\). **Graph Explanation:** - The graph displays \(H(j\omega)\), the frequency response of the LTI system. - The x-axis represents the angular frequency \(\omega\), ranging from \(-2\) to \(2\). - The y-axis shows the magnitude of \(H(j\omega)\). - The graph is a rectangular function with: - A height (magnitude) of 2. - Nonzero values only within the range \(-2 \leq \omega \leq 2\). **Given Input Signal:** \[x(t) = \cos(4t)\] **Objective:** Determine the output signal \(y(t)\) of the LTI system. **Answer Choices:** 1. \( y(t) = 0 \) 2. \( y(t) = \cos(4t)/2 \) 3. \( y(t) = \cos(4t) \) 4. \( y(t) = 2\cos(4t) \) **Understanding the Frequency Response:** The input signal \(x(t) = \cos(4t)\) can be expressed in the frequency domain, involving frequencies \(\pm 4\). Evaluating \(H(j\omega)\) at these frequencies (\(\omega = 4\) and \(\omega = -4\)): - From the graph, \(H(j\omega)\) is zero outside the range \(-2 \leq \omega \leq 2\). - Therefore, the input frequencies \(\pm 4\) are outside the effective range of \(H(j\omega)\). - The system will attenuate these frequencies to zero. **Conclusion:** Given the frequency response and the provided input signal, the system output \(y(t) = 0\). Thus, the correct answer is: \[ \boxed{y(t) = 0} \]
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