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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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sampleMidterm3hr_sol - ECE 108 Signals and Systems Spring...

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