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P-3.5 Make a sketch of the spectrum (vs. ω) of the signal defined by: 10128 3 x(t) = Σ k=-3 1 1 + jk ejkt Label each spectrum line with the corresponding complex amplitude in polar form.

P-3.5 Make a sketch of the spectrum (vs. ω) of the signal defined by: 10128 3 x(t) = Σ k=-3 1 1 + jk ejkt Label each spectrum line with the corresponding complex amplitude in polar form.

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-3.5 with details on how to do it. Thank you. 

### Signal Processing Exercises #### P-3.4 **Given Signal:** \( x(t) = \sin^3(27\pi t) \) 1. **(a) Determine a formula for \( x(t) \) as the real part of a sum of complex exponentials.** 2. **(b) Determine the fundamental period for \( x(t) \).** 3. **(c) Plot the spectrum for \( x(t) \).** #### P-3.5 **Task:** Make a sketch of the spectrum (\( \omega \)) of the signal defined by: \[ x(t) = \sum_{k=-3}^{3} \frac{1}{1 + jk} e^{jkt} \] **Instructions:** - Label each spectrum line with the corresponding complex amplitude in polar form. #### P-3.6 **Description:** Shown in Fig. P-3.6 is a spectrum plot for the periodic signal \( x(t) \). The frequency axis has units of rad/s. --- The exercises above guide the analysis of complex signals through the determination of their representations, periods, and spectra. Clear steps provide a structured approach to solving such problems, enabling a deep understanding of the temporal and spectral characteristics of the given signals.
Transcribed Image Text:### Signal Processing Exercises #### P-3.4 **Given Signal:** \( x(t) = \sin^3(27\pi t) \) 1. **(a) Determine a formula for \( x(t) \) as the real part of a sum of complex exponentials.** 2. **(b) Determine the fundamental period for \( x(t) \).** 3. **(c) Plot the spectrum for \( x(t) \).** #### P-3.5 **Task:** Make a sketch of the spectrum (\( \omega \)) of the signal defined by: \[ x(t) = \sum_{k=-3}^{3} \frac{1}{1 + jk} e^{jkt} \] **Instructions:** - Label each spectrum line with the corresponding complex amplitude in polar form. #### P-3.6 **Description:** Shown in Fig. P-3.6 is a spectrum plot for the periodic signal \( x(t) \). The frequency axis has units of rad/s. --- The exercises above guide the analysis of complex signals through the determination of their representations, periods, and spectra. Clear steps provide a structured approach to solving such problems, enabling a deep understanding of the temporal and spectral characteristics of the given signals.
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