Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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Fourier Transform
In mathematics, the Fourier transformation is a mathematical transformation that rotates responsibilities by using region or time into tasks depending on the local or temporal frequency, such as the rendering of a musical track in step with existing volu…
Question

Draw the time domain signals for which the Fourier Transform is given as follows. 

**Transcription of the Image for Educational Purposes** In this section, we explore the given mathematical expression related to signal processing: \[ b)\ X(\omega) = \pi [\delta(\omega - 1000) + \delta(\omega + 1000)] \] **Explanation of the Expression:** 1. **X(ω):** This represents a signal in the frequency domain, often used in signal processing to analyze the frequency components of a signal. 2. **π [δ(ω - 1000) + δ(ω + 1000)]:** This expression contains two Dirac delta functions, which are used to represent impulses at specific frequencies. - **δ(ω - 1000):** This delta function creates an impulse at 1000 rad/s in the positive frequency domain. - **δ(ω + 1000):** This delta function creates an impulse at -1000 rad/s in the negative frequency domain. - The factor of π scales the amplitude of these impulses. **Interpretation:** This signal, X(ω), consists of two impulses symmetrically positioned about the origin in the frequency domain at ±1000 rad/s. This can represent a signal that has significant frequency components at these two frequencies. The presence of such delta functions often simplifies analysis involving transforms like the Fourier Transform, due to their properties.
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Transcribed Image Text:**Transcription of the Image for Educational Purposes** In this section, we explore the given mathematical expression related to signal processing: \[ b)\ X(\omega) = \pi [\delta(\omega - 1000) + \delta(\omega + 1000)] \] **Explanation of the Expression:** 1. **X(ω):** This represents a signal in the frequency domain, often used in signal processing to analyze the frequency components of a signal. 2. **π [δ(ω - 1000) + δ(ω + 1000)]:** This expression contains two Dirac delta functions, which are used to represent impulses at specific frequencies. - **δ(ω - 1000):** This delta function creates an impulse at 1000 rad/s in the positive frequency domain. - **δ(ω + 1000):** This delta function creates an impulse at -1000 rad/s in the negative frequency domain. - The factor of π scales the amplitude of these impulses. **Interpretation:** This signal, X(ω), consists of two impulses symmetrically positioned about the origin in the frequency domain at ±1000 rad/s. This can represent a signal that has significant frequency components at these two frequencies. The presence of such delta functions often simplifies analysis involving transforms like the Fourier Transform, due to their properties.
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