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# E_TutQuestion - E TUTORIAL QUESTIONS E.1 Periodicity Energy...

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E T UTORIAL Q UESTIONS E.1 Periodicity, Energy and Power 1 Convert the following complex numbers into polar form, that is, in the form of . θ ρ j e s = (a) 4 3 j s + = (b) ( ) ( ) ( ) t jb t a t s + = (c) ( ) + = 4 10 cos 5 π π t t s 223

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224 Tutorial Questions 2 Convert the following complex numbers into rectangular form, that is, in the form of ω σ j s + = . (a) 6 2 j e s = (b) 3 2 k j ke s π = (c) where Re denotes the real part of a complex number. ( ) ( ) [ 5 20 5 Re + = t j e t s ] 3 Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period. (a) ( ) ( ) ( ) t t t x 6 sin 4 cos + = (b) ( ) ( ) t t t x 2 sin cos + = (c) ( ) ( ) [ ] 1 2 = t j e t x π 4 Determine whether the following signals are energy signals, power signals or neither. (a) ( ) ( ) 0 , > = a t u e t x at (b) , where u is the unit step function ( ) ( ) t tu t x = ( ) t (c) ( ) ( ) θ ω + = t a t x 0 cos 5 This is an optional question. The delta function ( ) t δ below is a “generalized” function and is frequently used. Find its energy. function impulse or Delta t 1 0 1 area ( ) t δ 0 t 0
Tutorial Questions 225 6 This is an optional question. The process of modulation below is frequently used to “up convert” the frequency of a signal to a higher value so that it can be more efficiently transmitted. Are the energy of ( ) t x and ( ) t y the same? ( ) Hz 5 3 to t x ( ) ( ) Hz 105 103 100 2 to t x e t y t j π = Modulation sinusoid Hz 100 100 2 t e π j E.2 Fourier Series 1 Determine the complex exponential Fourier series representation for the following. (a) ( ) ( ) t t x 5 sin 7 = (b) ( ) ( ) ( ) t t t x 6 sin 4 cos + = (c) ( ) t t x 2 sin = 2 (a) The signal ( ) ( ) t x t x = is real, symmetric about 0 = t and has a period of 0 0 2 ω π = T . Show that its complex Fourier series ( ) L L + + + + = t j t j t j e X e X e X t x 0 0 0 1 1 0 0 1 1 ω ω ω degenerates to become a cosine series ( ) ( ) ( ) L t a t a a t x 0 2 0 1 0 2 cos 2 1 cos 2 ω ω + + = where , , L are real. 0 a 1 a

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226 Tutorial Questions (b) A signal that is sometimes used in communications systems is the raised cosine pulse. The following figure shows a signal ( ) t x that is a periodic sequence of these pulses with equal spacing between them. Show that the first three terms in the Fourier series expansion of ( ) t x are given by ( ) ( ) ( ) L + + + = t t t x π π π 2 cos 2 1 cos 3 8 2 1 ( ) 5 . 0 2 cos 1 + t t , π ( ) t x 2 0 2 t 3 Show that the Fourier series coefficients for the signal ( ) t x shown below are given by = 2 sin 2 cos 1 2 π π π k k k X k 0 2 2 ( ) t x 1 t Hence, show that ( ) ( ) = = = = L L L L L L , 10 , 8 , 6 , 4 , 2 , 0 , , 0 , 11 , 7 , 3 , , 2 , 9 , 5 1 , , 2 k k k k k X k π π Then, show that the Fourier series expansion for the signal is ( ) ( ) [ ] t m m t x m m π π 1 2 cos 1 2 ) 1 ( 4 0 + + = =
Tutorial Questions 227 4 (a) This is an optional question. See that the two signals below are actually the same. Differentiate the signal. Sketch and express your result using delta or impulse functions.

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E_TutQuestion - E TUTORIAL QUESTIONS E.1 Periodicity Energy...

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