ch02 - Chapter 2 Signal and Linear System Theory 2.1...

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Chapter 2 Signal and Linear System Theory 2.1 Problem Solutions Problem 2.1 For the single-sided spectra, write the signal in terms of cosines: x ( t )=1 0 c o s ( 4 π t + π / 8) + 6 sin(8 π t +3 π / 4) =1 0 c o s ( 4 π t + π / 8) + 6 cos(8 π t π / 4 π / 2) 0 c o s ( 4 π t + π / 8) + 6 cos(8 π t + π / 4) For the double-sided spectra, write the signal in terms of complex exponentials using Euler’s theorem: x ( t )=5 e x p [ ( 4 π t + π / 8)] + 5 exp[ j (4 π t + π / 8)] +3 exp[ j (8 π t π / 4)] + 3 exp[ j (8 π t π / 4)] The two sets of spectra are plotted in Figures 2.1 and 2.2. Problem 2.2 The result is x ( t )=4 e j (8 π t + π / 2) +4 e j (8 π t + π / 2) +2 e j (4 π t π / 4) e j (4 π t π / 4) =8 c o s ( 8 π t + π / 2) + 4 cos (4 π t π / 4) = 8sin(8 π t )+4cos(4 π t π / 4) 1
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2 CHAPTER 2. SIGNAL AND LINEAR SYSTEM THEORY f , Hz f , Hz 0 2 4 6 0 2 4 6 10 5 π /4 π /8 Single-sided amplitude Single-sided phase, rad. Figure 2.1: Problem 2.3 (a) Not periodic. (b) Periodic. To f nd the period, note that 6 π 2 π = n 1 f 0 and 20 π 2 π = n 2 f 0 Therefore 10 3 = n 2 n 1 Hence, take n 1 =3 , n 2 =10 , and f 0 =1 Hz. (c) Periodic. Using a similar procedure as used in (b), we f nd that n 1 =2 , n 2 =7 , and f 0 Hz. (d) Periodic. Using a similar procedure as used in (b), we f nd that n 1 , n 2 ,n 3 =11 , and f 0 Hz. Problem 2.4 (a) The single-sided amplitude spectrum consists of a single line of height 5 at frequency 6 Hz, and the phase spectrum consists of a single line of height - π / 6 radians at frequency 6 Hz. The double-sided amplitude spectrum consists of lines of height 2.5 at frequencies of 6 and -6 Hz, and the double-sided phase spectrum consists of a line of height - π / 6 radians at frequency 6 Hz and a line of height π / 6 at frequency -6 radians Hz. (b) Write the signal as x b ( t )=3cos(12 π t π / 2) + 4 cos(16 π t ) From this it is seen that the single-sided amplitude spectrum consists of lines of height 3 and 4 at frequencies 6 and 8 Hz, respectively, and the single-sided phase spectrum consists
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2.1. PROBLEM SOLUTIONS 3 f , Hz 0 2 4 6 π /4 π /8 Double-sided phase, rad. f , Hz -6 -4 -2 0 2 4 6 5 Double-sided amplitude - π /8 - π/4 -6 -4 -2 Figure 2.2:
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4 CHAPTER 2. SIGNAL AND LINEAR SYSTEM THEORY of a line of height - π / 2 radians at frequency 6 Hz. The double-sided amplitude spectrum consists of lines of height 1.5 and 2 at frequencies of 6 and 8 Hz, respectively, and lines of height 1.5 and 2 at frequencies -6 and -8 Hz, respectively. The double-sided phase spectrum consists of a line of height - π / 2 radians at frequency 6 Hz and a line of height π / 2 radians at frequency -6 Hz. Problem 2.5 (a) This function has area Area = Z −∞ ² 1 · sin( π t/² ) ( π t/² ) ¸ 2 dt = Z −∞ · sin( π u ) ( π u ) ¸ 2 du =1 A sketch shows that no matter how small ² is, the area is still 1. With ² 0 , the central lobe of the function becomes narrower and higher. Thus, in the limit, it approximates a delta function.
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ch02 - Chapter 2 Signal and Linear System Theory 2.1...

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