[Solutions Manual] Fourier and Laplace Transform - Antwoorden

[Solutions Manual] Fourier and Laplace Transform - Antwoorden

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Answers to selected exercises for chapter 1 Apply cos( + ) = cos cos - sin sin , then 1.1 f 1 ( t ) + f 2 ( t ) = A 1 cos t cos 1- A 1 sin t sin 1 + A 2 cos t cos 2- A 2 sin t sin 2 = ( A 1 cos 1 + A 2 cos 2 ) cos t- ( A 1 sin 1 + A 2 sin 2 ) sin t = C 1 cos t- C 2 sin t, where C 1 = A 1 cos 1 + A 2 cos 2 and C 2 = A 1 sin 1 + A 2 sin 2 . Put A = p C 2 1 + C 2 2 and take such that cos = C 1 /A and sin = C 2 /A (this is possible since ( C 1 /A ) 2 +( C 2 /A ) 2 = 1). Now f 1 ( t )+ f 2 ( t ) = A (cos t cos - sin t sin ) = A cos( t + ). Put c 1 = A 1 e i 1 and c 2 = A 2 e i 2 , then f 1 ( t ) + f 2 ( t ) = ( c 1 + c 2 ) e it . Let 1.2 c = c 1 + c 2 , then f 1 ( t ) + f 2 ( t ) = ce it . The signal f 1 ( t ) + f 2 ( t ) is again a time-harmonic signal with amplitude | c | and initial phase arg c . The power P is given by 1.5 P = 2 Z /- / A 2 cos 2 ( t + ) dt = A 2 4 Z /- / (1 + cos(2 t + 2 )) dt = A 2 2 . The energy-content is E = R e- 2 t dt = 1 2 . 1.6 The power P is given by 1.7 P = 1 4 3 X n =0 | cos( n/ 2) | 2 = 1 2 . The energy-content is E = P n =0 e- 2 n , which is a geometric series with 1.8 sum 1 / (1- e- 2 ). a If u ( t ) is real, then the integral, and so y ( t ), is also real. 1.9 b Since Z u ( ) d Z | u ( ) | d, it follows from the boundedness of u ( t ), so | u ( ) | K for some constant K , that y ( t ) is also bounded. c The linearity follows immediately from the linearity of integration. The time-invariance follows from the substitution = - t in the integral R t t- 1 u ( - t ) d representing the response to u ( t- t ). d Calculating R t t- 1 cos( ) d gives the following response: (sin( t )- sin( t- )) / = 2 sin( / 2) cos( t- / 2) / . e Calculating R t t- 1 sin( ) d gives the following response: (- cos( t ) + cos( t- )) / = 2 sin( / 2) sin( t- / 2) / . f From the response to cos( t ) in d it follows that the amplitude response is | 2 sin( / 2) / | . g From the response to cos( t ) in d it follows that the phase response is- / 2 if 2 sin( / 2) / 0 and- / 2 + if 2 sin( / 2) / < 0. From 1 2 Answers to selected exercises for chapter 1 phase and amplitude response the frequency response follows: H ( ) = 2 sin( / 2) e- i/ 2 / . a The frequency response of the cascade system is H 1 ( ) H 2 ( ), since the 1.11 reponse to e it is first H 1 ( ) e it and then H 1 ( ) H 2 ( ) e it . b The amplitude response is | H 1 ( ) H 2 ( ) | = A 1 ( ) A 2 ( )....
View Full Document

This note was uploaded on 01/22/2009 for the course ME 3322 taught by Professor Neitzel during the Spring '07 term at Georgia Institute of Technology.

Page1 / 87

[Solutions Manual] Fourier and Laplace Transform - Antwoorden

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online