EECS-360 Signal and System AnalysisSpring Semester 2005

Exam #2: 18 April 2005

NAME: k E- '/ KUID:

GENERAL INSTRUCTIONS

. Put your name on each page, in case the pages get separated.

. There are 100 points possible on this exam.. Show Your Work! Partial credit awarded for correct set-up.

. Show all work for each problem on the exam pages. I have tried to leave plenty of space for you toshow your work, and the back side of the last page is blank for you to use. Please do not use any otherpaper unless absolutely necessary. If you do use the back side of the last page or other paper, clearlyindicate on the problem page that there is additional work elsewhere.. Clearly indicate your answer to each part (unqerline, box, etc.).

. When numerical values are requested, give your answers as decimal numbers (not fractions); that is,perform all calculations rather than leaving your answer as some complicated expression,. Whenever units are appropriate for your answers, provide them. Points will be deducted for failure to do

this.. The following abbreviations will be used throughout this exam:

CT Continuous-Time 1

DT Discrete-Time j

LTI Linear, Time-InvariantC- D Continuous-to- DiscreteD-C Discrete-to-ContinuousFIR. Finite Impulse ResponseIIR Il)finite Impl1lse Response

. The following relationships may be useful to you:

ej8 = cos 8 + j sin 8

The Z-transform of anu[n] is -~ 1 for laz-l l< 1.1-az-

Spring 2005

EECS-360 Exam #2 - 2 - Name: k £- j

1. (10 points) An FIR filter is "sandwiched" between an ideal sampling circuit (i.e., C-D converter) ('and an ideal reconstruction circuit (i.e., D-C converter), both using a sampling frequency of 18 t$kHz. Design the FIR filter so that the overall system will null out any CT sinusoid with frequency

r ~3 kHz. Your final answer must be a block diagram with decimal numbers for all multiplieri D values. A

(;)f) ::. o?rrfo =- {pofj/),TT rls ~ ~o:::. A -:.. .!::2!!!-£ - .J[ "ts (~" IT - .3 rdd «(1 AJ ~

jOt ~+ ~ ~ere;S 6{/( (~) d i -: e..~J 13

1//7\- (I jrr/3 -( \ /1 -j1T'/3 -I ) - I ( J'rr/3 -j1T/3\ -( 7-2-tt l C-) - - e l) l - e Z: - - e f-t ) t + z::

, " --"'

o? (' fY;) 7r/3 ::::1

tJOf tt(2-)~ /-t-'+l-2-

tJ~ ~ In]:: X[/l]- ,x[II-I] + X [n -2]

)( Lnl ~ [rtJ .

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EECS-360 Exam #2 - 3 - Name: i{ 6. i-- M -:.. '" .

2. The frequency response of a particular sixCorder FIR filter has a linear phase function. This filterhas a zero at Z = 1. 5e-j3;r/5 and another zero at z = ej;r/4.

a. (5 points) Find all of the other zero values for this FIR fliter.- IL 1 .. . I r ,f I" 'I ~ /-L-

IA~c.((' PNJJ)(; T lo i~ a C e~1 ~ 6D 11 ~) if.1) /J.1y;;;f 2-t:>~

A ( , r I ... - j 31T1s- - J j37r/S-pp u. lAJ (~ ll> - 1,5 e -=]) ~ - ().l..91 e .,<f-

.J DI'31i S

lo*-:: !,6--eJ ..1--L . -J' 371"15"- &--~~

It -=. D,f.J:,t e

A rr [~ t4J't~ ~ 0 :: eJ rr(1 ~ i fJ = ij rr(~ ':;. l() *d--

.-L - jir/1-- ~lo-h- - e -~D

--,-- -\

4 o}4r l::e NJ 0 ~ .. 37Tj..)- -J' rr:1<jJ~3rrls .. 31r~- - J t'

~-~~~~--- ~s f. J , ~) c=-~b. (5 points) Describe in your own words the advantage of a filter that has a frequency response

with a linear phase function. ,

LIA'~r pkJe if\. .tl-.(rvt~tj (j i~rU{ ~~~ /.c /{ .JiIY/! -Shift

('n +{~. III fIJ""{lc/)..~r 0. f({/..()r- f)J(U /lNdr- f/~'()Je +-~ a t(~ ~j~tWf f':~pOf)5~, (In ~ paJ:ha ~ i1J[Jrlb "

~{st&l't 511~ ~ (1 e;rJ( ~~ W!UIA. Ih ftJ.J 16£ .

7r:J jJ, I J X (t-) I { ~.s -e. f\.~ /\~ IA\,U (f\ fh pfJJJ.b ~ ~It-) ~ k X Ll -t6)) /)J~ ;~ j~ ~ 6eJ-tJ tl rJ .

~~e -5h{~kd J~.f'j(,~ tJt xii-)) tU~J.. (~ ~ JejJ!'QJPro~. Petr t{5 CJ. d (5 ~f (( M( ~JJ trO/YVJ ~ r ft\4 {( M I Spring 2005 i

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a bo':' (3. A particular IlR fj~ has ~ differen~e equatiQn..ef-c b f ':. - L

/""" / ~y[n] = (-0.4)y[n-l] +x[n] -2x[n-l]

. a. (3 points) Find the system function H(z) for this IlR filter.r / b -I

B f. r III ) bD +- ,I:)1.{ {~p4C,"t-lM if't)fY\ ;t-(l::::. ~ -/_-1.J O,l

I/- ( l ).:. _l-=2-~-~

I..f-OI1~-'

b. (3 points) Draw the Direct Form I block diagram for this IlR filter.

.x [ n 1 ~ [1\ 1--.

Prof. P~tr Spring 2005

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EECS-360 Exam #2 '.;!!d:;~ - 5 - Name: -k £ 1

c. (3 points) State the specific initial rest conditions that would have to be imposed (orassumed) jn order to find the~impul~e !!:2J!~or this IIR filter using recursion and the

r::~ff~r~n~k;ati;~ p/WvOl,.Q ) X [n. ] =- Ii L J"1.]) {)J ~ I J i.!> D {o r n ~ n 0 ~ LJ .

.50 I uJ'i! t;J~ 11.( J D ~ ( ~ r d!J U;rf'l .e j ,Lk-f

h [n] ~ D /l ~ f!o ::.[) .(h Lvi] il ~ sp-eclhL~ Lt1] ~~ X [1\] -::. t)[J1] J

d. (11 points) Invoking the initial rest conditions that you just specified, use recursion and thedifference equation to find the impulse response h[n] of this IIR filter. Specify h[n] for allvalues of n.

~[IlJ::: -D.4fLLf>../] +- &LfI] -J(5'[/I-/J (IIJ~ xLltl:--t)[it]) '.-' -~- -~ ~ L ] ~(~ e.-J h [ i1 J )

1£ )' 1\1 =- 0 n ..( D LI A& ~tJJ 1'e..r1-JrLL D 1. ()

tL [c; J:: - D, ~ h;f -I] +- D to, ] - ~ 0- I, j ~ 1 i

I .fJ1: I ~D ~ I ! I

n[t}==- -tJ,~ kiD] +DllJ -0< (J/[D] ::.. -o.4-J ::- -:IlL! i

'j I '. 0'0 .

h[zI=- -()tcf h[r] -1--6/1-] -J~!t] ";; -lJ,1f;,'-() ~ ()tqt.e0 ~ 0

~ [31=- -tJ,c.fhlz-] -t--dj31-:<'it<J ~ -O,4(tJIQI.'-) = -tJ,3ffL(

f\ ofe +i.t pa fie t'/l: J:~ r /l~ J) h [t\J ~ -'C/,i{ A [I\-(]

::;0.' - ~"- h [it 1 =- 0 /1 .:::- 0

tt [0] -=- I

It u]:: -;,1 Dr h [n]:: -J,L/( -011)",-1 n? IfL- .I

J\ [n1 ~ -;(4 (-tJrLf) or h[1l1 ~-~(-()Iq)'l ==- loto.1jrt

(l~1

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4. (10 points) This problem concerns the reduced Direct Form II block diagram below. I1..

x[n] y[n]

I

fIJ

Using only relationships and equations taken directly from the block diagram, demonstrate that thedifference equation corresponding to this block diagram is:

y[n] = al y[n - 1] + bo x[n] + bi x[n - 1]

~ [(1.1= 10 W [/\ 1 +b./ w [11-/1

50 4 [n-Jl::. boWLI\-l] +- hI W [h-Z].)

A~f)1 W [1\1::: al W [1\-IJ + X[~J

50 w[n-I]::, q, W [A-Z} t-X[/\-I]

Pv-t .t+ ~~~tliDf' ~

~ I}1}:' bo [ Q I If) [fl- U + X [11] S + b t fIX 1 IAJ L f\ -2.1 +- X [f\ -I}~ I -

::. b6a, W ["-IJ +- b I ~l vJ [h-z.} .f-b~ x[nl.+-bt .x [A-(1

~ L)\ 3 ::. ~ \ j L f\ -IJ + box Ln.] +.. b \ X LY\ - I Jb~1

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c""..C i IEECS-360 Exam #2 -7 - Name: k£ "i -~_.~~~

5. (5 points) A certain DT system has a difference equation of the form:

y[n] = aly[n -1] + azy[n - 2] + box[n] + b1x[n -1] + bzx[n - 2]

With specific values for some of the difference equation coefficients, this filter be designed to havepoles at z = O. 8ej3n/4 and z = O. 8e-j3n/4. Give the coefficients and their spe~ific numerical values

that will accomplish this.1-' e. I I 1~~~~~R. ' ~ r pO (PA 0) r C) ()) J(d. k NJ (Jj 'f--N..7{-

C-il :;. - r 2. (j'~ 0{ l :: ; r ClIj @-

PeN) r':. (),2' ~ !;)::. 31r)i

51) Cfz ::.; - (Dlff-) L:;;G~~~~~. oJ ~ I ~ 0:< C ~ ..g-J tdIJ (3 rrf tiJ ':: ~I ill j{-- 0 I t /) .:. f=~J

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---: EECS-360 Exam #2 - S - Name:

6. The system function H(z) of a particular DT system is:

9 -2-zH(z) = (1- O. SZ-l )(1 + O. 5Z-1)

a. (5 points) Find the difference equation for this DT system.1+ {i:-)::. ___1_= 2 - 2 :::... __~-=.i~~ .

I ( 0. I . \ -I I J - '2- -I ,1 - '?..+ -O/O"IOIS)l: - O,,"t: l-b.3c -0'7C-

8 ( ~ D1c..1~~ 'V\. +tioOII\I\ If [~),:- __k~_=~.!.~- 't-~6 -~~~~4 r /_11 2 -( ~-L.j . '1 ( '{: - Clz.- c--

". .",.

b. (5 points) Find the poles and zeros of this DT system and sketch them on a z-plane pole:'zerodiagram. J '

HLt?:-)~ (3 -t-1 (3 -f-t-1 -~ ltf'OS IJJr l::.. :t ~. ~-t>1 ,("l:j_I,LO,5"~ fo r~ ~ ?::; (),2'

"-"'"'-,._- ~--~ v.tJ t ~ - () f s:

) N\ f 2:-) .() (\'l -+ ~ I t'('...lt

~ ~,

R ~")7g e., ( 1.';)

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Spring 2005

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c. (15 points) Use z-transform techniques to find the impulse response h[nJ for this DT systemfrom the system function H(z). Do not use recursion and the difference equation to findh[nl. .

Nume~r ~ teAonl ~ so.AIY1~ ()rdtr~ d'l/(de OJ..~ -/- cl, ..:) {p ,Sf tJ , .f-S i

/- ()(3~-I_O(1l~--q-=l--=-~- 60 /.f [l):: ~15+ ~~~j-I -2-;ItS- -0 I tSt:: - 1:

~-.

-/

(.p,54-0tt-5t ,P()t'i~d _f~~ rlpdt11(~ "

.Ii / ) L 1 H t t.--~~,-If Lt-- ::. el, Q T' --I .-;:-~;'= I ""T" C"v'" ('"' ~ ... I- VIl$' C {t~ ,

A ::::.. .f!!ci.i!!.~~'! '"=; ~~i!!.d:?-:t.[ = if (S-- ~1+ () /5"2- (c. ..~ t> (~ I + 0 I {r; 2 S

( ~! : /,). ~ )

. B:: !!!.~:!:~¥ '/ =- ~5=~.~-::.. I, q z

/-0 (8"t c:: -D.S- 1-1-1, &("t-'~-..z) . 1+-

~I U.!C1 'fie htlSr( 2-tf'CilYbti) rm po rr .. ~ JS P"1e/

/L[111:. t?sf[i1] -I is-P'(/)'~JI\L(Lnl +- /,Q2 L-O's)~ [/1]'" '

- Prof. Petr - Spring 2005

---~ EECS-360 Exam #2 - 10 - Name:

7. The impulse response h(t) of a certain CT LTI system is shown below. Note that h(t) = t for-1 < t < 0 and 0 elsewhere.

: h(t)I

-3 -2 -1 I 1 2 3I

-+--I

a. (3 points) Is the system represented by h(t) causal? You must give a specific reason for youranswer.

&f: ttUJ ~~: C~ulo.J: req u. ( t.I/) h l &J ::- 0 t <: 0 )

Df...li- ~ n {~).:::. t- -I L. I:: ~O .

b. (3 points) Is the system represented by h(t) bounded-input, bounded-output (BIBO) stable?You must give a specific reason for your answer. r 00

15 8(80 skb(e: lIJi{( .be 'j}) IhLtildt <cxO- --- -I()

t!el'.Ql rIA(~)I,::; 1z.

c. (4 points) Give an expression for and sketch the output y(t) of this system for an input ofx(t) = 30(t - 2).

b ~ L T.I) /)} ~ X (~)::: 3 f ( t -L) ) j (I:-) -::: .3 h (f - 2- ) )

60 ~ [[-):: 3{ t. "'-2..) +0 r -1 ~ i-l. .c.::. () {I,t.e! +6 r 1<' t L ~) ,

=- () ~{~e~ ~ ~ I

{ ~ [I::-)

-2- -I f , ~ .3-- - i

.,.

( t'I\; I

.,, i

-3 -.:.- 1

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EECS-360 Exam #2 - 11 - Name: T'\ \- 1 I

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V ~d. (10 points) Find the outputy(t) of this system for the input given below. You need not sketch

y(t). You may continue your work on the back of this page if necessary.

: x(t)I

1

- - - - -1- - --

I-3 -2 -1 I 1 2 3 t

;X (/:: -A) Dt'1 A-()~;l (t/ip ~ s(cde),

~---J---l t- t >.

t: .( -I: fib D f/'!f'{o.p / ~ l) ~ (I:-) ::-- D t..c: - (

, '--I ~ t ~ 0 , ~ (f) ~ f t: A d ~ ~ {A L J t ":;. t (t 2..- (-I J '- )

- ( .-5 0 j~::::Li7t~r~=E~~~

D ~ 1: .<: l. " , D f 0 ?- ~ (t-) .::- t ~ d ~ ~ f I\L _::. i- CD - [-I) )

( Sc ~)-:~7~ --oZ~~~--/~1 ~ , , "~~~-_o - .

1~~_~3 ~ ~ (f-)'::::' ~ D A Jil ::.c. i A2-1 0 ::-- t l{) - (f-2-)l.)

't.-J- t:-L

.';0 [ (6-j:: - i It:-z.j 'Z.. /~ tL~3 .,-,-" ~ " ~

t: ~ j \ / --,.~, ,"",-_."-:"7

;;.' I\~ () rr-L) t..c \(1 fJ -- , "'- fL V~'-f./'-r' ,..,.

so! 4 (~.) ::;.. 0 t: ~)",.l-.~-- i

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