The discrete-time Fourier transform (DTFT) X(w) = Σ=-∞x[n]e¯jwn of a sequence x[n] is displayed below for 0≤ @ < 2. X(w) + π/2 1 0.5 π 3π/2 a) Is x[n] periodic? Why or why not? If periodic, what is its period? b) Is x[n] real-valued? Why or why not? n=-00 2π 3 2 c) Use Parseval's Relation to find the total energy Σx[n]² in the sequence x[n].
The discrete-time Fourier transform (DTFT) X(w) = Σ=-∞x[n]e¯jwn of a sequence x[n] is displayed below for 0≤ @ < 2. X(w) + π/2 1 0.5 π 3π/2 a) Is x[n] periodic? Why or why not? If periodic, what is its period? b) Is x[n] real-valued? Why or why not? n=-00 2π 3 2 c) Use Parseval's Relation to find the total energy Σx[n]² in the sequence x[n].
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Author:Robert L. Boylestad
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![The discrete-time Fourier transform (DTFT) \( X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \) of a sequence \( x[n] \) is displayed below for \( 0 \leq \omega < 2\pi \).
**Graph of \( X(\omega) \):**
The graph represents \( X(\omega) \) with the following details:
- The x-axis is labeled \( \omega \) and ranges from 0 to \( 2\pi \).
- The y-axis represents the magnitude of \( X(\omega) \).
- There are significant points marked at \( \omega = \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
- The graph consists of rectangular steps with the following values:
- From 0 to \(\pi\), the magnitude is 1.
- From \(\pi\) to \(2\pi\), the magnitude is 0.5.
**Questions:**
a) Is \( x[n] \) periodic? Why or why not? If periodic, what is its period?
b) Is \( x[n] \) real-valued? Why or why not?
c) Use Parseval’s Relation to find the total energy \( \sum_{n=-\infty}^{\infty} |x[n]|^2 \) in the sequence \( x[n] \).
d) Sketch \(|Q(\omega)| = |\sum_{n=-\infty}^{\infty} q[n] e^{-j\omega n}| \) if \( q[n] = x[n - 3] \).
e) The discrete-time Fourier transform \( Y(\omega) = \sum_{n=-\infty}^{\infty} y[n] e^{-j\omega n} \) of another sequence \( y[n] \) is displayed below for \( 0 \leq \omega < 2\pi \). Using the Frequency Shift property of the DTFT, derive a relationship between \( x[n] \) and \( y[n] \).
**Graph of \( Y(\omega) \):**
- The x-axis is labeled \( \omega \) and ranges from 0 to \( 2\pi \).
- The y-axis represents the magnitude of \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b207c38-3ae5-4958-b6f8-8e524b4be30c%2Fa6151138-b3b0-4b0f-b172-640d390885e7%2Fcvpvzxu_processed.png&w=3840&q=75)
Transcribed Image Text:The discrete-time Fourier transform (DTFT) \( X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \) of a sequence \( x[n] \) is displayed below for \( 0 \leq \omega < 2\pi \).
**Graph of \( X(\omega) \):**
The graph represents \( X(\omega) \) with the following details:
- The x-axis is labeled \( \omega \) and ranges from 0 to \( 2\pi \).
- The y-axis represents the magnitude of \( X(\omega) \).
- There are significant points marked at \( \omega = \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
- The graph consists of rectangular steps with the following values:
- From 0 to \(\pi\), the magnitude is 1.
- From \(\pi\) to \(2\pi\), the magnitude is 0.5.
**Questions:**
a) Is \( x[n] \) periodic? Why or why not? If periodic, what is its period?
b) Is \( x[n] \) real-valued? Why or why not?
c) Use Parseval’s Relation to find the total energy \( \sum_{n=-\infty}^{\infty} |x[n]|^2 \) in the sequence \( x[n] \).
d) Sketch \(|Q(\omega)| = |\sum_{n=-\infty}^{\infty} q[n] e^{-j\omega n}| \) if \( q[n] = x[n - 3] \).
e) The discrete-time Fourier transform \( Y(\omega) = \sum_{n=-\infty}^{\infty} y[n] e^{-j\omega n} \) of another sequence \( y[n] \) is displayed below for \( 0 \leq \omega < 2\pi \). Using the Frequency Shift property of the DTFT, derive a relationship between \( x[n] \) and \( y[n] \).
**Graph of \( Y(\omega) \):**
- The x-axis is labeled \( \omega \) and ranges from 0 to \( 2\pi \).
- The y-axis represents the magnitude of \(
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![d) Sketch |Q(w)| = |Σn=-oq[n]e-jun | if q [n] = x[n - 3].
=-a
e) The discrete-time Fourier transform Y(w) = Σ - y[n]e-jun of another sequence y[n] is
displayed below for 0 <<2. Using the Frequency Shift property of the DTFT, derive a
relationship between x[n] and y[n].
Y(w)
1
0.5
π/2
π
3π/2
2π
W](https://content.bartleby.com/qna-images/question/c23eb98a-2dff-445f-a3f1-279aad19fd90/4681c3b4-064a-4e28-8673-40a714cab73d/7nvgj6_processed.png)
Transcribed Image Text:d) Sketch |Q(w)| = |Σn=-oq[n]e-jun | if q [n] = x[n - 3].
=-a
e) The discrete-time Fourier transform Y(w) = Σ - y[n]e-jun of another sequence y[n] is
displayed below for 0 <<2. Using the Frequency Shift property of the DTFT, derive a
relationship between x[n] and y[n].
Y(w)
1
0.5
π/2
π
3π/2
2π
W
![d) Sketch |Q(w)| = |Σn=-oq[n]e-jun | if q [n] = x[n - 3].
=-a
e) The discrete-time Fourier transform Y(w) = Σ - y[n]e-jun of another sequence y[n] is
displayed below for 0 <<2. Using the Frequency Shift property of the DTFT, derive a
relationship between x[n] and y[n].
Y(w)
1
0.5
π/2
π
3π/2
2π
W](https://content.bartleby.com/qna-images/question/c23eb98a-2dff-445f-a3f1-279aad19fd90/4681c3b4-064a-4e28-8673-40a714cab73d/i74rlyn_processed.png)
Transcribed Image Text:d) Sketch |Q(w)| = |Σn=-oq[n]e-jun | if q [n] = x[n - 3].
=-a
e) The discrete-time Fourier transform Y(w) = Σ - y[n]e-jun of another sequence y[n] is
displayed below for 0 <<2. Using the Frequency Shift property of the DTFT, derive a
relationship between x[n] and y[n].
Y(w)
1
0.5
π/2
π
3π/2
2π
W
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