Midterm#1

# Midterm#1 - Department of Electrical and Computer Engineering The Ohio State University WINTER 2008 ECE 700 Digital Signal Processing 11:30 AM-l:18

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Unformatted text preview: Department of Electrical and Computer Engineering The Ohio State University WINTER 2008 ECE 700 Digital Signal Processing 11:30 AM-l :18 PM 2/08/08 Midterm Examination Directions 0 You are allowed to have two sheets of paper with anything you want on it. No other books or notes are allowed in the examination. 0 The examination has 5 questions and 12 questions. 0 Write clearly and show all work for full credit. 0 If you are making any assumptions in solving any of the problems, please state these clearly at the beginning of your solution. 0 The ECE Honor System applies to this examination. At the end of the examination, if the pledge below applies, please sign your name in the line below the pledge. “No aid given, received or observed during the course of this examination.” Signature Your namei VH7ka T qul ma @ 1. Consider the system shown below. C/D is an ideal continuous-to-discrete converter and D/C is an ideal discrete-to-continuous converter. Let i=3X104 Hz and i=10“ Hz. 7; T2 a. Sketch Yd(ej“’). Clearly label all relevant frequencies and amplitudes. b. Sketch YC(jQ). Clearly label all relevant frequencies and amplitudes. -27rX9X103 o 2nX9X103 3(91") 1 m’6 LIE-Ml)? 7g? 0 a) Wag): Xce)‘HC?) K377) 9'77 {Tiff This a e is intentionall left blank' continue Problem 1 V WPEE“ 21(th w: We): go «w; 5‘?) H (wmmgﬁ) : 3' we“) H (am x (xi/92%) H a»: law ><(w;é)ag«;§:')) N ’1 up]: 3? Va J, . 3 1T \gui'ﬂ' ~E2: , ,3: is 2. Sketch the overall frequency response of the system shown below. G(z) is an ideal low pass ﬁlter with a cut-off frequency of n/4. Va): X6?) 6C?) w 69): mama") 24+ m) We): 6&2”) (Xe) 6,6295% Mal) : we) me“) + m) moééwp" ZMM 3. Consider the system shown below. (X XW wfn] m a. Express W (em) and Y (e’”)in terms of X (e’o’). b. If X (e’”) is as shown below, sketch Y (em). c. Suppose that x[n] is obtained by sampling a continuous time signal at 1000 Hz. What is the effective sampling rate of y[n]? X(e~’") 1 This a e is intentionall left blank' use for Problem 3 C) L mom (woo-3:) :(500 U< <3 4. Consider the system shown below. d. Write an expression for V(z) in terms of X 1 (z) and X2(z). e. Write down y1 [n] and y2[n] in terms of x1[n] and x2[n]. Simplify your answer as much as possible. ‘ . -1 : J:(><,(z)zz‘z+><262) f X, Lela ‘+ Xaégl) :: LXCZ 'i ><Z(2)) :. X169)2_i+XZ/CLZ) 3 v: (2): Z" (X, a“) 2‘31 M2?) : X; C2‘)2‘Z+ >91?) 2" (L63): 1 09:6?)1‘ WW“) .1 ‘ “l . ‘ Caz-inf maz- 105(2)?) -1 5. Consider the two channel ﬁlterbank shown below. Let a. Derive Y (z) in terms of X(z) and the ﬁlter transfer functions. b. Show that the ﬁlterbank is alias free if H1(z) = Ho(—Z)> Go(z) = 110(2), 61(2) = H1 c. Under the alias-free conditions above, what is the transfer ﬁmction of the overall system in terms of H0 (2) ? a) y, (2): W2) W2) nahXQWN”?' w, a) : i (V. (23+ V. (“39) l : i (xcéwoéﬁﬂXéa‘) H06?» I I J _._J, J J_ ‘1 W162): ﬁ<><L%‘)H,(E")Z_L we?) Mag») 62 “16%): «12,422 ) 3 i (Name) + xezmez) ) - \J (XL?) HL;)_X(—z)H,(~2)) 27) l x! . ~ (“BJHICzEj‘L l/IZéBL (x9) Hl (2)2 >< ~l \(u w- 6062))? w ~/ (:62): iIC%(3)H0[Z)+ vaElHob?))’ 46(2) 3 lot? :Ulil’é’léz) ‘ a. a ): 12CX{})Hu£e)—><C'})H.C‘Z)lZ ’- (m5? ) L l ’— waPL?) / I l XL?) Maisoﬁ/oéxénwb )6m(3)+ H'- ‘J 10 L xcz)H:(e)6z,m—y(~z) (GENHQU 3’ This a e is intentional] left blank' use for Problem 5 In Wk We)»: Em) yea] «Q W {M I @oéz) £0]:[H°éz) Hivw] [67»(2) o : H062)! \$002M magma Jig: H,CZ):H°(~2) @042) am?) 5:62): Hm: H003) W0 '2} (1700?” It?) 6"! Lg) ; mm + HoW'Z) : O n ‘ HQ) H093] [140(2) 9) (j (2)3:DUE) ><L~2)] ‘: Hoc~z)~Ho(2) HaC-Z) , 0263 {‘HOZC'E) :7; EX (3) VOW] LH )0 j 1-: machcg) +HJ(-2)) [40(3) H162) J W Hog?) m6?) W?) 11 ...
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## This note was uploaded on 10/25/2009 for the course ECE 700 taught by Professor Un during the Spring '09 term at Ohio State.

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Midterm#1 - Department of Electrical and Computer Engineering The Ohio State University WINTER 2008 ECE 700 Digital Signal Processing 11:30 AM-l:18

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