Name: Suden Number: Page EE.35: Specrum Analysis and Discree-Time Sysems MIDTERM EXAM, :3PM 4:3PM, November 4, 4 (closed book) Examiner: Ha H. Nguyen Noe: There are four quesions. All quesions are of equal value bu no necessarily of equal difficuly. Par marks for each quesion are indicaed. Full marks shall only be given o soluions ha are properly explained and jusified.. (Signal Transformaions) [6] (a) A coninuous ime signal x() is shown below. Nealy skech each of he following signals: (i) x( ), (ii) x( + 4), (iii) x(3 ), (iv) x ( ). x().5.5 9 8 7 6 5 3 4 5 6 7 8 9 x( ).5.5 9 8 7 6 5 3 4 5 6 7 8 9 x(+4).5.5 9 8 7 6 5 3 4 5 6 7 8 9 x().5.5 9 8 7 6 5 3 4 5 6 7 8 9 x(3 ).5.5 9 8 7 6 5 3 4 5 6 7 8 9 x( /).5.5 9 8 7 6 5 3 4 5 6 7 8 9 EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan
Name: Suden Number: Page [3] (b) Consider he same coninuous-ime signal as in par (a). Deermine and skech is even and odd pars. Noe: The provided emplaes migh be useful. You do no have o use all of hem if you do no need o. x().5.5 9 8 7 6 5 3 4 5 6 7 8 9.5.5 9 8 7 6 5 3 4 5 6 7 8 9 x () e.5.5 9 8 7 6 5 3 4 5 6 7 8 9 x().5.5 9 8 7 6 5 3 4 5 6 7 8 9.5.5 9 8 7 6 5 3 4 5 6 7 8 9 x () o.5.5 9 8 7 6 5 3 4 5 6 7 8 9 [] (c) Is x() an energy or power signal? Why? EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan
Name: Suden Number: Page 3. (Convoluion) [5] (a) Consider a discree-ime LTI sysem wih impulse response h[n] and inpu x[n] as shown below. Find and nealy skech he oupu y[n]. Hin: The oupu can be found by using he Lineariy and Time-Invariance properies insead of performing he convoluion sum. h[n] 3 4 5 6 n x[ n] 3 4 5 6 n x( ) h( ) EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan
Name: Suden Number: Page 4 [5] (b) Consider a coninuous-ime LTI sysem wih impulse response h() = e u() and inpu x() as shown below. Find and roughly skech he oupu y(). x( ) h( ) EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan
Name: Suden Number: Page 5 3. (Properies of Fourier Series Coefficiens) The following figures show he magniude and phase specra of hree coninuous-ime periodic signals. Answer he following quesions: [4] (a) For each signal, deermine wheher he signal is real or complex. Explain your answer. [] (b) Which signal is a real-valued and even funcion? Which signal is a real-valued and odd funcion? Explain your answer. Magniude (vols) Phase (rad) (i) 3 4 3 4 Magniude (vols) Phase (rad) (ii) 3 4 3 4 Magniude (vols) Phase (rad) (iii) 3 4 3 4 [4] (c) For he signal corresponding o he specra in (i), do he following: Wrie a Fourier series equaion for he ime waveform x(). Wha is he dc componen of his signal? Find he average power of his signal. EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan
Name: Suden Number: Page 6 4. (Fourier Series Represenaion) One echnique for building a dc power supply is o ake an ac signal and full-wave recify i. Tha is, if he inpu o he full-wave recifier is he ac signal x(), hen is oupu is y() = x(). [3] (a) Consider he inpu signal x() = cos() shown in he figure below. Nealy skech he oupu y(). Wha are he fundamenal periods of x() and y()? Inpu x()=cos().5.5 5 5.5.5 Oupu y()= cos() 5 5 (sec) [5] (b) Find he rigonomeric Fourier series coefficiens B k and C k of he oupu y(). [] (c) Wha are he ampliudes of he dc componens of he inpu and oupu signals, respecively? EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan
Name: Suden Number: Page 7 Poenially Useful Facs: Even and odd pars: x e () = [x() + x( )], x o() = [x() x( )] Convoluion sum: x[n] h[n] = h[n] x[n] = x[k]h[n k] = h[k]x[n k] Convoluion inegral: x() h() = h() x() = x(τ)h( τ)dτ = FS represenaion of CT periodic signals (CTFS): Exponenial: x() = a k e jk h(τ)x( τ)dτ Ampliude-Phase: x() = a + A k cos(k + θ k ) k= Trigonomeric: x() = a + [B k cos(k) C k sin(k)] k= a A k e jθ k = B k + jc k A a k = Bk + C k ( ) θ a an Ck B R{a k } = A k cos(θ k ) C I{a k } = A k sin(θ k ) a x()e T T jk d B x() cos(k)d, C x() sin(k)d T T T T Remark: The ampliude-phase and rigonomeric forms only apply for real-valued signals. The complex-conjugae symmery of he Fourier series coefficiens of real-valued periodic signals: a a k, a a k B k Parseval s relaion: T T x() d = a k Ideniies: e j e j = cos() + j sin() = cos() j sin() cos(x) cos(y) = [cos(x + y) + cos(x y)] Indefinie Inegrals: e a d = ea (a ) a sin(b)e a d = cos(b)e a d = cos(a)d = a sin(a) sin(a)d = a cos(a) ea a + b[a sin(b) b cos(b)] ea a + b[a cos(b) + b sin(b)] EE.35: Specrum Analysis and Discree-Time Sysems, Universiy of Saskachewan