Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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### Transcription for Educational Website

**A continuous-time signal \( x_a(t) \) has a continuous-time Fourier transform \( X_a(\Omega) = \int_{-\infty}^{\infty} x_a(t) e^{-j\Omega t} dt \) shown below:**

#### Graph Description:
- The graph is a single triangle centered at the origin.
- The x-axis is labeled \( \Omega \) (in rad/sec).
- The y-axis shows \( X_a(\Omega) \).
- The peak of the triangle is at \( \Omega = 0 \) with a maximum value of 1.
- The base of the triangle extends from \( \Omega = -100\pi \) to \( \Omega = 100\pi \).

**Tasks:**

a) Suppose \( x_a(t) \) is sampled at 150 samples/sec resulting in the discrete-time signal \( x[n] \).

   - **Task:** Sketch the discrete-time Fourier transform (DTFT) \( X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \). Label all important values on your sketch.

b) Repeat part a) with a sampling rate of 75 samples/sec.
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Transcribed Image Text:### Transcription for Educational Website **A continuous-time signal \( x_a(t) \) has a continuous-time Fourier transform \( X_a(\Omega) = \int_{-\infty}^{\infty} x_a(t) e^{-j\Omega t} dt \) shown below:** #### Graph Description: - The graph is a single triangle centered at the origin. - The x-axis is labeled \( \Omega \) (in rad/sec). - The y-axis shows \( X_a(\Omega) \). - The peak of the triangle is at \( \Omega = 0 \) with a maximum value of 1. - The base of the triangle extends from \( \Omega = -100\pi \) to \( \Omega = 100\pi \). **Tasks:** a) Suppose \( x_a(t) \) is sampled at 150 samples/sec resulting in the discrete-time signal \( x[n] \). - **Task:** Sketch the discrete-time Fourier transform (DTFT) \( X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \). Label all important values on your sketch. b) Repeat part a) with a sampling rate of 75 samples/sec.
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