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Question 2. Consider that the impulse response of a system is given as: h[n] = [2()" ()"]u[n] If x[n] = (u[n] is applied to this system as input, calculate the output of the system USING FOURIER TRANSFORM.

Question 2. Consider that the impulse response of a system is given as: h[n] = [2()" ()"]u[n] If x[n] = (u[n] is applied to this system as input, calculate the output of the system USING FOURIER TRANSFORM.

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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OUTPUT USING FT( NEED NEAT HANDWRITTEN SOLUTION ONLY OTHERWISE DOWNVOTE).

**Question 2.** Consider that the impulse response of a system is given as: \[ h[n] = \left[ 2 \left( \frac{1}{2} \right)^n - \left( \frac{1}{4} \right)^n \right] u[n] \] If \( x[n] = \left( \frac{1}{2} \right)^n u[n] \) is applied to this system as input, calculate the output of the system USING FOURIER TRANSFORM. --- In this image, a discrete-time signal processing problem is proposed. The impulse response \( h[n] \) of a system is provided, along with a specific input signal \( x[n] \). The task is to calculate the output of the system when the given input is applied, using the Fourier Transform. - **Impulse Response:** \( h[n] = \left[ 2 \left( \frac{1}{2} \right)^n - \left( \frac{1}{4} \right)^n \right] u[n] \) This represents a linear combination of two scaled exponential terms, multiplied by the unit step function \( u[n] \), which ensures that the response is defined for \( n \geq 0 \). - **Input Signal:** \( x[n] = \left( \frac{1}{2} \right)^n u[n] \) This is a simple exponentially decaying sequence, also multiplied by the unit step function \( u[n] \), implying it is active for \( n \geq 0 \). The required operation involves using the Fourier Transform properties to find the system's output in the frequency domain. The output can be computed by finding the product of the Fourier Transforms of the input and the impulse response in the frequency domain and then applying the inverse Fourier Transform to obtain the time-domain output.
Transcribed Image Text:**Question 2.** Consider that the impulse response of a system is given as: \[ h[n] = \left[ 2 \left( \frac{1}{2} \right)^n - \left( \frac{1}{4} \right)^n \right] u[n] \] If \( x[n] = \left( \frac{1}{2} \right)^n u[n] \) is applied to this system as input, calculate the output of the system USING FOURIER TRANSFORM. --- In this image, a discrete-time signal processing problem is proposed. The impulse response \( h[n] \) of a system is provided, along with a specific input signal \( x[n] \). The task is to calculate the output of the system when the given input is applied, using the Fourier Transform. - **Impulse Response:** \( h[n] = \left[ 2 \left( \frac{1}{2} \right)^n - \left( \frac{1}{4} \right)^n \right] u[n] \) This represents a linear combination of two scaled exponential terms, multiplied by the unit step function \( u[n] \), which ensures that the response is defined for \( n \geq 0 \). - **Input Signal:** \( x[n] = \left( \frac{1}{2} \right)^n u[n] \) This is a simple exponentially decaying sequence, also multiplied by the unit step function \( u[n] \), implying it is active for \( n \geq 0 \). The required operation involves using the Fourier Transform properties to find the system's output in the frequency domain. The output can be computed by finding the product of the Fourier Transforms of the input and the impulse response in the frequency domain and then applying the inverse Fourier Transform to obtain the time-domain output.
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