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Assume the following input signal and initial conditions: x(n) = (-1)"u(n), y(-1) = 1, y(-2) = -1 a) Write the (linear constant coefficient) difference equation that corresponds to this system (the final LCCDE should only contain current/delayed time points of input and output signals).

Assume the following input signal and initial conditions: x(n) = (-1)"u(n), y(-1) = 1, y(-2) = -1 a) Write the (linear constant coefficient) difference equation that corresponds to this system (the final LCCDE should only contain current/delayed time points of input and output signals).

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
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**Block Diagram of an LTI System** The provided illustration represents a block diagram of a Linear Time-Invariant (LTI) system. The system is defined by its response to an input signal \( x(n) \) and produces an output signal \( y(n) \). **Diagram Explanation:** - **Inputs and Outputs:** - The signal \( x(n) \) enters the system. - The signal \( y(n) \) is the output from the system. - **Components:** - **Adders**: Indicated by circles with a plus sign, they combine incoming signals. - **Multipliers**: Indicated by arrows with values (2, -1), these components scale the incoming signals by the given factors. - **Delays**: Represented by \( z^{-1} \) blocks, they delay the signal by one time unit. - **Flow:** 1. The input \( x(n) \) passes through an adder. 2. Part of the signal branches off, entering a delay block \( z^{-1} \). 3. The delayed signal is multiplied by -1. 4. Another branch of the delayed signal goes through a delay and is multiplied by 2. 5. The results of the multiplications are summed and scaled by another factor of 2 before producing the output \( y(n) \). **Assumptions and Initial Conditions:** - Input signal: \( x(n) = (-1)^n u(n) \) - Initial conditions: \( y(-1) = 1 \) and \( y(-2) = -1 \) **Tasks:** a) **Difference Equation:** - Derive the linear constant coefficient difference equation (LCCDE) using the given block diagram. The equation should only include the current and delayed values of input and output signals. b) **Particular Solution:** - Determine the particular solution of the derived equation from part a). c) **Zero-State and Zero-Input Solutions:** - Find the zero-state and zero-input solutions for the equation from part a). This content provides a fundamental understanding of constructing and solving equations for systems represented by block diagrams in signal processing contexts.
Transcribed Image Text:**Block Diagram of an LTI System** The provided illustration represents a block diagram of a Linear Time-Invariant (LTI) system. The system is defined by its response to an input signal \( x(n) \) and produces an output signal \( y(n) \). **Diagram Explanation:** - **Inputs and Outputs:** - The signal \( x(n) \) enters the system. - The signal \( y(n) \) is the output from the system. - **Components:** - **Adders**: Indicated by circles with a plus sign, they combine incoming signals. - **Multipliers**: Indicated by arrows with values (2, -1), these components scale the incoming signals by the given factors. - **Delays**: Represented by \( z^{-1} \) blocks, they delay the signal by one time unit. - **Flow:** 1. The input \( x(n) \) passes through an adder. 2. Part of the signal branches off, entering a delay block \( z^{-1} \). 3. The delayed signal is multiplied by -1. 4. Another branch of the delayed signal goes through a delay and is multiplied by 2. 5. The results of the multiplications are summed and scaled by another factor of 2 before producing the output \( y(n) \). **Assumptions and Initial Conditions:** - Input signal: \( x(n) = (-1)^n u(n) \) - Initial conditions: \( y(-1) = 1 \) and \( y(-2) = -1 \) **Tasks:** a) **Difference Equation:** - Derive the linear constant coefficient difference equation (LCCDE) using the given block diagram. The equation should only include the current and delayed values of input and output signals. b) **Particular Solution:** - Determine the particular solution of the derived equation from part a). c) **Zero-State and Zero-Input Solutions:** - Find the zero-state and zero-input solutions for the equation from part a). This content provides a fundamental understanding of constructing and solving equations for systems represented by block diagrams in signal processing contexts.
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