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(a) Write an equation for x (t) as a sum of cosines. (b) Is x (t) a periodic signal? If so, determine its fundamental period and its fundamental frequency. (c) Explain why negative frequencies are needed in the spectrum.

(a) Write an equation for x (t) as a sum of cosines. (b) Is x (t) a periodic signal? If so, determine its fundamental period and its fundamental frequency. (c) Explain why negative frequencies are needed in the spectrum.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Please assist with p-3.1 a,b,c with details on how to do it. Thank you. 

# Problems The problems in this section address the major concepts of this chapter. On the Companion Website, there are many related homework problems which can be used to review and practice. Most of them have detailed solutions. ## P-3.1 A signal \( x(t) \) has the two-sided spectrum representation shown in Fig. P-3.1. ### Diagram Explanation (Figure P-3.1) The figure is a two-sided spectrum representation of a signal \( x(t) \) showing several spectral components at different frequencies. The horizontal axis represents the frequency \( f \), ranging from -175 to 175. The vertical axis represents the amplitude of the spectral components. Specific components are marked: - At \( f = \pm 50 \), the spectral component is \( 7e^{i\pi/3} \). - At \( f = \pm 175 \), the spectral component is \( 4e^{-i\pi/2} \). - At \( f = 0 \), the spectral component is 11. ### Tasks 1. **Write an equation for \( x(t) \) as a sum of cosines.** 2. **Is \( x(t) \) a periodic signal? If so, determine its fundamental period and its fundamental frequency.** 3. **Explain why negative frequencies are needed in the spectrum.** ## P-3.2 A signal composed of sinusoids is given by the equation \[ x(t) = 10 \cos (800\pi t + \pi/4) + 7 \cos (1200\pi t - \pi/3) - 3 \cos (1600\pi t) \] ### Tasks 1. **Sketch the spectrum of this signal, indicating the complex amplitude of each frequency component. Make separate plots for real/imaginary or magnitude/phase of the complex amplitudes at each frequency.** 2. **Is \( x(t) \) periodic? If so, what is the period?**
Transcribed Image Text:# Problems The problems in this section address the major concepts of this chapter. On the Companion Website, there are many related homework problems which can be used to review and practice. Most of them have detailed solutions. ## P-3.1 A signal \( x(t) \) has the two-sided spectrum representation shown in Fig. P-3.1. ### Diagram Explanation (Figure P-3.1) The figure is a two-sided spectrum representation of a signal \( x(t) \) showing several spectral components at different frequencies. The horizontal axis represents the frequency \( f \), ranging from -175 to 175. The vertical axis represents the amplitude of the spectral components. Specific components are marked: - At \( f = \pm 50 \), the spectral component is \( 7e^{i\pi/3} \). - At \( f = \pm 175 \), the spectral component is \( 4e^{-i\pi/2} \). - At \( f = 0 \), the spectral component is 11. ### Tasks 1. **Write an equation for \( x(t) \) as a sum of cosines.** 2. **Is \( x(t) \) a periodic signal? If so, determine its fundamental period and its fundamental frequency.** 3. **Explain why negative frequencies are needed in the spectrum.** ## P-3.2 A signal composed of sinusoids is given by the equation \[ x(t) = 10 \cos (800\pi t + \pi/4) + 7 \cos (1200\pi t - \pi/3) - 3 \cos (1600\pi t) \] ### Tasks 1. **Sketch the spectrum of this signal, indicating the complex amplitude of each frequency component. Make separate plots for real/imaginary or magnitude/phase of the complex amplitudes at each frequency.** 2. **Is \( x(t) \) periodic? If so, what is the period?**
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