2. Consider a discrete-time LTI system described by the following time-domain relationship, where \( x[n] \) is the system input and \( y[n] \) is the system output.\[ y[n] = x[n] + \frac{1}{2} \big( x[n-1] + x[n+1] \big) \]a) Is the system causal? Why or why not?b) Find the frequency response of the system \( H(\omega) = \frac{Y(\omega)}{X(\omega)} \).c) Plot the frequency response \( H(\omega) \) over the frequency range \( -2\pi \leq \omega \leq 2\pi \). Is the frequency response periodic in frequency?d) Find the output signal \( y[n] \) if the input signal is \( x[n] = e^{j\pi n} \).

2. Consider a discrete-time LTI system described by the following time-domain relationship, where x[n] is the system input and y[n] is the system output. 1 y[n] = x[n] + 7 (x[n − 1] + x[n + 1]) a) Is the system causal? Why or why not? b) Find the frequency response of the system H(w) c) Plot the frequency response H(w) over the frequency range -2 ≤ w ≤ 2π. Is the frequency response periodic in frequency? d) Find the output signal y[n] if the input signal is x[n] = ejân = Y(w) X(w)

2. Consider a discrete-time LTI system described by the following time-domain relationship, where x[n] is the system input and y[n] is the system output. 1 y[n] = x[n] + 7 (x[n − 1] + x[n + 1]) a) Is the system causal? Why or why not? b) Find the frequency response of the system H(w) c) Plot the frequency response H(w) over the frequency range -2 ≤ w ≤ 2π. Is the frequency response periodic in frequency? d) Find the output signal y[n] if the input signal is x[n] = ejân = Y(w) X(w)

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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Question

2. Consider a discrete-time LTI system described by the following time-domain relationship, where \( x[n] \) is the system input and \( y[n] \) is the system output.

\[ y[n] = x[n] + \frac{1}{2} \big( x[n-1] + x[n+1] \big) \]

a) Is the system causal? Why or why not?

b) Find the frequency response of the system \( H(\omega) = \frac{Y(\omega)}{X(\omega)} \).

c) Plot the frequency response \( H(\omega) \) over the frequency range \( -2\pi \leq \omega \leq 2\pi \). Is the frequency response periodic in frequency?

d) Find the output signal \( y[n] \) if the input signal is \( x[n] = e^{j\pi n} \).

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