Type:
Solution Manual
Resource:
Principles of Communications
Edition:
6th Edition
Author(s):
Rodger E. Ziemer William H. Tranter
Chapter 2
Signal and Linear System Theory 2.1
Problem Solutions
Problem 2.1 a. For the single-sided spectra, write the signal as x(t) = 10 cos(4πt + π/8) + 6 sin(8πt + 3π/4) = 10 cos(4πt + π/8) + 6 cos(8πt + 3π/4 − π/2)
= 10 cos(4πt + π/8) + 6 cos(8πt + π/4) h i = Re 10ej(4πt+π/8) + 6ej(8πt+π/4)
For the double-sided spectra, write the signal in terms of complex exponentials using Euler’s theorem: x(t) = 5 exp[j(4πt + π/8)] + 5 exp[−j(4πt + π/8)] +3 exp[j(8πt + 3π/4)] + 3 exp[−j(8πt + 3π/4)] The spectra are plotted in Fig. 2.1. b. Write the given signal as i h xb (t) = Re 8ej(2πt+π/3) + 4ej(6πt+π/4) to plot the single-sided spectra. For the double-side spectra, write it as xb (t) = 4ej(2πt+π/3) + 4e−j(2πt+π/3) + 2ej(6πt+π/4) + 2e−j(6πt+π/4) The spectra are plotted in Fig. 2.2. 1
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CHAPTER 2. SIGNAL AND LINEAR SYSTEM THEORY
Figure 2.1:
c. Change the sines to cosines by subtracting π/2 from their arguments to get xc (t) = 2 cos (4πt + π/8 − π/2) + 12 cos (10πt − π/2) = 2 cos (4πt − 3π/8) + 12 cos (10πt − π/2) i h = Re 2ej(4πt−3π/8) + 12ej(10πt−π/2)
= ej(4πt−3π/8) + e−j(4πt−3π/8) + 6ej(10πt−π/2) + 6e−j(10πt−π/2)
Spectral plots are given in Fig. 2.3.
2.1. PROBLEM SOLUTIONS
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Figure 2.2:
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CHAPTER 2. SIGNAL AND LINEAR SYSTEM THEORY
Figure 2.3: