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

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

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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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.
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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