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1 EECS 20N: Structure and Interpretation of Signals and Systems MIDTERM 2 Department of Electrical Engineering and Computer Sciences 02 November 2005 UNIVERSITY OF CALIFORNIA BERKELEY LAST Name FIRST Name Lab Time A". (10 Points) Print your name and lab time in legible, block lettering in the appropriate spaces provided above AND on the back of the last page.. This midterm should take you up to 80 minutes to complete. You will be given at least 80 minutes-up to a maximum of 90 minutes-to work on the midterm.. Aside from TWO two-sided 8.5 x ll-inch formula sheet for your exclusive use, you may not access, or cause to be used or accessed, any reference in print or electronic form at any time during the quiz. Computing, communication, and other electronic devices (except dedicated timekeepers) must be turned off. Noncompliance with these or other instructions from the teaching staff-including, for example, commencing work prematurely or continuing beyond the announced stop time-is a serious violation of the Code of Student Conduct.. The midterm printout consists of pages numbered 1 through 10. When you are prompted by the teaching staff to begin work, verify that your copy of the midterm is free of printing anomalies and contains all of the ten numbered pages. If you find a defect in your copy, notify the staff immediately.. Please write neatly and legibly, because ifwe can't read it, we can'tgradeit.. For each problem, limit your work to the space provided specifically for that problem. No other work will be consideredin evaluating your midterm. No exceptions.. You must clearly and convincingly explain all your answers..we hope you wrestle and beat this midterm to the ground! 1 --

2 MT2.1 (20 Points) The impulse response 9 of a discrete-time, linear, time-invariant (LTI) system G : [Z -t JR]-t [Z -t JR]is given by: \InEZ, g:z-tjr I if n ;:::0 g(n) = { o ifn<o. Determine the unit-step response s of the system G by plotting it and labeling all its salient features. That is, determine and plot the output of the system in response to the unit-step input signal u, where \In E Z, t>t:ltl _ l ifn;:::o un( ) -. { o ifn<o 1\~ 0 tl -1 ~ l ";> ~ y~ dt~-it)~o " k. >- '" ~ ~ ~ S(tt):~ 'K!)b)::: L ~t\<.)atr\-k)= L. 1 = (\+\ 1<:-00 k: 0 I'\~O V\< 0 ~b/c.. Q(q::o \11«0 G-r..thic...\1~:lA~;~d i\,e "e<\-o" ~;e'" of- co~~ol.itq"., ~('\) ~ d-(w\)-t'd(yt-\') T ~V1-~)TH. Of -::.'-'.(1'\)-t"'-lV\-\) 3(i\) + ~ (V\-o1.) -r... 0' d., 2

3 MT2.2 (20 points) A discrete-time system F : [Z --+ JR.]--+ [Z --+ JR.]is known to be time invariant. Three exemplary input-output behaviors (Xi, Yi), i = 1,2,3, of the system are depicted in the figure below (all signals are zero outside the regions shown): x (1) -1 F JL (2) Yln) 012 n o 1 n x (n) (3) n -1 F -th (-1) 'l-1f~ 4 n ~ Yln) (2) n (a) Could the system F be causal? Explain your reasoning succinctly, but clearly and convincingly. \J "'-~ \ J 'b..t <t. tt\t\j* 0 fqr,,=-\ (whe<,e:.~ t\:- \) (b) Could the system F be memoryless? Explain your reasoning succinctly, but clearly and convincingly. \ ~o'. R '5o-~t~W"\.~~t ~ s v\ot C 0\ "'-'!Io-..\ C o..v\o\q-t \'<:. Me. m.0cj ~ ~S. ~e<:jl ~c..t: ~e.w\o(a\e.~~ --='"> C~'-\.,:>J.. J..?~~~ ;~ \o~\'c.~~ ur~~~~'\ttq.. NoT ~"'~ } -3? No' }J\eMQ('~esS.

4 (c) Determine the output of the system F in response to the unit-impulse signal 0 (Kronecker delta function), where o:z-tjr \:InE Z O(n) = I if n = 0., { 0 ifn#o r- ~~ T,m~ ~~,,~('~X.. \1e"'-te) \<.~o~'\"'d-\rt ~li\)-:x3(~t~) W't\~~o'\~~~t ih~ ('~'S.fo>r\seot t\;e. scjillvv\.to ~e. "\,,,,'It \m, ~..\~~ ; <:> ([ (~))(1'\') ::. ds(\'\;-\r ') ~ (PWll) -t 0 \ (d) Could the system F be linear? Explain your reasoning succinctly, but clearly and convincingly. Not \;",e,d\r \ ~lt'\ ') -=. )1.,(",') - x~ly\;-'). _. I 4-1\ A. f\ -'- f IS \ T -=--'> f c; (,~~\Of\S~ \0 ~)5.t. X.;I.(f\)-=-x.;t(I\t1) I~ 'd.;tj'5.\. O;l.t"l~1;a.("t\)., f\s~"'-w\.e. f; C:. \~Y\~o..r. ~e..v"\ d(t'\) - d-:0;-\') " <,. f \otic\, ~ ~\~" U -, 0 \ ~ ~ \~'S. ~"=> t\qt ~e.. <s '"""'~ ;~ ~ OI./r (c.'). W~ ""0..\1~ ~ 0-.~ (f(5> ') ) (Y\ ') ~...t ~~.h~~l c.o~ l:ct; 0"" ~('e~<'~) F c:.~",,-v\q! b~ \~"'~'\ ("'. 4

5 MT2.3 (20 points) Consider a causal, discrete-time, SISO LTI system whose [A, B, C, D] state-space representation includes the following state-update equation: Sl(n + 1) = '-v-" '-v--"'--v-" ~ s(n + 1) A s(n) B ~ 'I Sl(n) + [ s2(n + 1) ] [ 0 2] [ s2(n) ] [ 0 1] x(n). The input signal and the state response are x : No -t IRand s : No ---t IR, respectively. Throughout this problem, let the input signal x be zero (Le., x(n) = 0, "in 2: 0), and consider the initial state given by (3a, (3E IR): s(o) = a [~] + (3 [~]. (a) Determine a simple expression for s(n), the state of the system at time n ("in> 1). ~ t Otf; I')" A$(:.1: ~1l~1 T ~I~~1L~} O«(~[~JrN;!)G] \. ~ [ ) 1 f\ \ ~ 1 s (~') -= ~ (~) 0 J T ~ (~) L3 J

6 MT2.4 (20 points) A real, causal, discrete-time linear time-invariant (LTI) system is characterized by the following delay-adder-gain block diagram. The input and output at time n are denoted by x(n) and y(n), respectively. Each block 0 corresponds to a delay by one sample; that is, if the input to the delay block D is a signal r, the output of the delay block is the signal q, where q(n) = r(n-l) for all n. s~ (V\t\) S (t\.or\) x(n),i,rdlsln), ~~S1(n) y(n) (a) Determine the linear, constant-coefficient difference equation that governs the input-output behavior of the system. d(~'): X(Y\)T~X("'-\) 1-"Sx.(V\-;;() (b) Determine and provide a well-labeled plot of the impulse response h of the system. ~((\ ') -::. ~(",') -r ~ ~(V\-\) -\-~ ~(V\-~ ') 3 I _~l' \,t'" ) O)~ (c) The outputs of the delay blocks are selected as the state variables sl(n) and s2(n), as shown in the figure. For this selection of state variables, determine the corresponding [A, B, C, D] state-space representation of the LTI system. "S\l"'''\''\) \Q 'l S\~~\) :'~~(Y\) l 1 -:::. \5(l~) 1 -T I Q ] Xl"'-) ~ $,2.(" 1'\) L Q 0 J ls.;l.~v\) ) s 0"'"\') -= X (0(\) '--.r--'-r-' ~ A ~ ()("') "" X ( )t- ~ 5.:!.l'\)-r 3. 5, (VI.) ;;;,al") -=-1> ~1\"5,(.)l,-,. x( ) ~ lsji,t!!)) l' 6 c.. D

7 MT2.5 (20 points) Consider a discrete-time signal x: Z -? JRhaving the following the properties:. x(n+4l) = x(n), 'VlE Z. 2 L x(n) = 2. n=-l 2 L (-ltx(n) = 4. n=-l 2 2 '"' 7r '"'. 7r. ~ x(n) cos(2'n) = ~ x(n) sm(2'n) = 0. n=-l n=-l Determine the complex exponential Fourier series coefficients X_l, Xo, Xl, and X2 for the signal x. From the coefficients, determine and provide a welllabeled plot for the signal x. The following complex exponential Fourier series expressions for a periodic discretetime signal having period p may be of potential use to you: x(n) = L Xk k=(p) eikwon

8 MT2.6 (20 points) This problem explores some of the properties of finite-state machines. Throughout, let the set D = {O,1, absent} denote an alphabet. Then each relevant input alphabet will be in D2. For every input pair (Xl(n), X2(n)) E D2, Xl(n) and X2(n) denote the top and bottom input symbols, respectively. (a) Consider the finite-state machine A shown below. y (i) Is machine A memoryless? Explain your reasoning succinctly, but clearly and convincingly. ~e.~. It Aoeos<'\ 'T M~r wh.=r ~\j~?l",-'ce ;~. ~ s tv\o.c.h~~e. ;s \'~~~~\\o.." is> ~ o~-s'ovte. N\~t..'h~",\:. (5ee. ~~('\ Llt')). (ii) Find a simpler deterministic state machine that is bisimilar to machine A. x, A '''-e.. o..-l\s..je.,r T. ~\\ ~<>..('t S~O\A(\~ (jv\.r 0..01\ ",:>-.Je., T"o )0..('1- (t-). 8

9 (b) Finite-state machine B is constructed feedback composition shown below: by placing machine A in the y B f (i),which of the following guard sets G1,G2 makes machine B wellformed? (I) Gl = {(I, On II Gl = {} { G2={(I,ln () { G2={(0,1),(I,ln Gl = {(I, 0), (1, 1), (0, In (III) Gl = {(I, 0),(0,on (IV) { G2= {} { G2 = {(O,on (.r) 13 is"" t' \\-yo~"",,-l bt <>"-""t 5-or -w. e.,,~ XI"- \ ) ~e-'e ~ Me. ;-..,0,,~,,-,t~<;d.f"t4 toi"t'.. l~)) em:.'): \) ISv.)~t\ -t Q('t'V\.E:.~\'~c."''''-~~ 50(' ~~~ 1'Ob.-t)\~"\t~I"'} i'\~ 4' )(,) ~~('~ ~s ~ "-"';I~~ f~)c.~.a~\;l1r) x,"::.-0 -'"> d~~:' \ Itrr\ 0 '1 r.. XI = \ ~ -\,:-')C,,-::. \ \...u-,.. x, ':. ~ <J~)(;I.:;' 'V ; x\-:.\~ (\-=-X~:.O. cr,.. l"n:) g,;, t-iwr. No "o""t,,"\\:e.~~"d-5-ix~ltoi.j- for XI'" O.. (ii) Assuming G1 and G2 are such that machine B is well-formed, can you conclude that B is memoryless? Explain your reasoning succinctly, but clearly and convincingly...l.: +4: \\ t.. \. t - \- I ' \ 4- (,_-IP r \l. '+: e ' :shc.. :S;I'\~~.'5I""'~ 1 B. IS "-Ie -1C1t'M.e.~, I 1'5 U'S'''''''~' \ 0 Q"'~ QJ 'h~ T9 o"'h""if,\el<: (\MII\ U 1V\~~\"\e.. S " : 6r~/O~} {a'15ft :&9

10 \ LASTName 5a\~ tqfi~ FIRST Name Lab Time Problem Points Your Score Name Total 130 J,O ;).0 O O O ;}..O 1-; 0 10