sampleMidterm3hr_sol

sampleMidterm3hr_sol - ECE 108 Signals and Systems Spring...

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Unformatted text preview: ECE 108 Signals and Systems Spring 2006, Tiffany Jing Li Additional Practice Problems for Midterm 2 Carefully go over the notes, the homework, the quizzes, and the problems in this sam- ple midterm.. Also make sure that you have a good understand of the fundamental concepts. in True or False: (a) If 32(t) is real, its Fourier Transform .X(w) o X (m) is aiways conjugate symmetric ( T ) o is always even( “T ) o AX(w)is always odd( "T ) (b)Thepmkfltfsnfl%%)mi”=IO/&( “r ) ii‘hna bate,‘ (e) A periodic signal has Fourier series, but it does not have Fourier transform" ( 22‘: (Cl) An aperiodic signal (—00 < t < oo) has Fourier transferm but does not have Fourier series. ( T ) (e) If a signal can be represented as a iinear sum (with either finite or countably infinite terms) of sinusoidal signals, then what can be concluded for this signal: / or, afm‘m’re o The signal must be periodie( )./ 1 n n o The signal must be real( T ) 7cm” a for"? “0’” W vfiflfll big I w i ‘ fli’. \ - 1 The signal must‘oe even( F, )Wtfly qurrzr‘SWfég ‘ o The signal must be odd( ..i ) k 2.‘ Simplify the following expressioas: (a) 5(w + 3) i, I w: -3 (It/o +3) 2«3} (b)(sin1(fw))6(w) : & mk<wfi kSEHCCkw) Eta) QM) :: )andoMrwr— e51“) v f (0).]:(iaiél5fle-fildt ZU3+¢) f in" S U-‘tHUf *2: fi 5 W i 3“ Consider“ a system specified by the equation (1)2 + 21:) +1)y(t) = 13m) (1) (a) Is this system asymptotically stable, marginally stable, or unstable? Arr—A2: -i s-labta 2 (b) Find the zerowinput response of this system fort 2 0, if the initial conditions are 210(0) = 2: and 316(9) m 1» WW}: Cialr (Si/c m Oates I, £45 é e ht (c) Find the unit impulse response of this system. bhzo Am:— mw {mele We) {Pm aw] (mt) ‘an “‘19: C. 6% + C32.'7(:.£3m1L ‘0’} My): “‘51 84+ awe”: m c3625 {nH'r‘aL wudr‘fim ;, 18“ (47:0) :1 O [én (ft :0) .2: I _ t CV1®L20 “*3 3 “C,+CZ:] :9 CILO ’CLZI I 11M):— {D‘[fe*~tj Jaw) 4a A certain periodic signal is given by f(t) m 3 -§~ x/Ecoswt) + sin(2t) + si11(3t) m 0,5 eos(5t + W/3) (a) What is the fundamental period? W0”; 1 ['72. 22;; 22.7; (b) Write the compact trigonometric Fourier series (Hint: combine the sine and cosine terms of the same frequency. All terms must appear in the cosine form with positive amplitudes. This can always be done by suitably adjusting the phase.) fewer} + 26:95 (gt-r3;- H‘fiw cos»: (it “$)+Q5ws(ngt+3§:fii) 5 ‘ 7)" 4‘ ‘ 7r ~—-~ Cr C: 2m (9; 2; 3w; T63 7F Jw 721; 3 C5 ((2) Sketch the trigonometric Fourier spectra 6022.3" 62:2 ’ C3: ’ CFZQIS... (92:7? (93 7i" “an 37C 0.“ cider; :2 0 5, Find the Fourior transform of the unit impukse function 6(t)‘ 95%)] 3)::8126) 2“ —.0c.0.t m -‘ f: “W 40:: C" Q / 'iLQX'fbaa k w £265 owfrt/pmr badge 243 , 36%” $3 6. Prove the Fourier transform pair ejwflt ~c:> 271—6 (11) — 'LUg) From W 1525+ pram/m: 50%) air; I Symmae-h’j C-h‘mvjmguemg dun/($1): F06) 6;? 2% J fizz/“3’17?” (SC—w) gfme'tfmwmgl r'nvgrsr'wm ' Saafr‘na, 13" £194) D ELI/wag}, \ 7 fie.) 7'. Prove the time convolutional property: if f1(t) 4m F101;) and fflt) <==> F2(UJ), then >E= 4: F1(1U)F2(’LU) "fey—l'lyaflk .255 ...
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This note was uploaded on 08/06/2008 for the course ECE 108 taught by Professor Li during the Spring '08 term at Lehigh University .

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sampleMidterm3hr_sol - ECE 108 Signals and Systems Spring...

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