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] + (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 -2π ≤ w ≤ 2π. Is the frequency response periodic in frequency? d) Find the output signal y[n] if the input signal is x [n] Y(w) X(w) = ejлn
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] + (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 -2π ≤ w ≤ 2π. Is the frequency response periodic in frequency? d) Find the output signal y[n] if the input signal is x [n] Y(w) X(w) = ejлn
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Please answer part D. Thank you!
![**Discrete-Time LTI System Analysis**
Consider a discrete-time Linear Time-Invariant (LTI) system described by the following time-domain relationship, where \( x[n] \) is the system input and \( y[n] \) is the system output.
\[ y[n] = x[n] + \frac{1}{2} (x[n-1] + x[n+1]) \]
Tasks:
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 \( x[n] = e^{j\pi n} \).**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b207c38-3ae5-4958-b6f8-8e524b4be30c%2F6d7383e6-1305-476b-93b3-c6e596116e7d%2F2me5m7_processed.png&w=3840&q=75)
Transcribed Image Text:**Discrete-Time LTI System Analysis**
Consider a discrete-time Linear Time-Invariant (LTI) system described by the following time-domain relationship, where \( x[n] \) is the system input and \( y[n] \) is the system output.
\[ y[n] = x[n] + \frac{1}{2} (x[n-1] + x[n+1]) \]
Tasks:
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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