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EECS-360 Signal and System AnalysisSpring Semester.2005

Exam #1: 7 March 2005

NAME: ~ f.. ~- KUill:

GENERAL INSTRUCTIONS

. Put your name on each page, in case the pages get separated.. There are 110 points possible on this exam.. Show Your Work! Partial credit awarded for correct set-up.

. Show all work on each problem on the exam pages. I have tried to leave plenty of room for you to showyour work, and there is an extra page at the end for you to use. Please do not use any other paper unlessabsolutely necessary. If you do use the extra page or other paper, clearly indicate on the problem page

that there is additional work elsewhere.. Clearly indicate your answer to each part (underline, box, etc.).. When numerical values are requested, give your answers as 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-TimeDT Discrete- Time

C-D Continuous-to-DiscreteD-C Discrete-to-Continuous

. The following relationship may be useful to you:

ej8 = cos {} + j sin{}

cos A. cos B = 0.5 (cos(A + B)+cos(A - B))

Prof. Petr Spring 2005

EECS-360 Exam #1 - 2 -

1. (5 points) This problem concerns the continuous-time signal x(t) shown below.I

-1- x(t)1 I

IIII t t 0 1 2 3 t

In the space below, sketch z(t) = 2x(-3t + 3). Be sure to carefully label all important points on boththe time axis and the amplitude axis. ~ - / / ))- J.~l-3L-l-J .

s-cJt o.,.,p ) -h- 7 \ ~-= ~ $'~{+-I- .~e. +{~e.

~ I/trf t (.0 (t\ pf'~.!' J'

, J,7-l-3t+3)J.

--, 43

C "ett: t:.. D i (o ):. ;Z.l<- (. D + ';) .:.. l X [3 )'""""-" ~

P '. I LA "-,../, J- L p6°,d w~l'( X (i) iU;"r1fS / k 0 VDIi\J' 1J),~r-.e. C"t~) JI.AVt'1p~ l> 10 ~ <1

2. (5 points) Prove the following. For any signal z(t), the signal w(t) = z(t) + z( -I) is an even functionoft.

£ 'J~ : W Lt:-) ~ /JJ ( -t-) .re r aiJ tlAJ (-' t:)~ ~ (-t) t. r- (-( ~t-)) : ~ L -t-) 4- l:-l &)

'::.. -t-(.f:.) +- t: (-I::-) -- ~ (6-)

~O lJJ L ~ t)~ W (i-\., C{ rJ A~~ l.J l t:: j if. tJ!...eM, .

)

Prof. Petr Spring 2005

EECS-360 Exam #1 - 3 - Name: k: ~ ~- Ib 0 . .1 ' 5'

I DOt) H:c: ~ ~ I<tOO ttr.- ~ I 0 0 . ;.., q/ r ~

3. Consider the following signal: x(t) = 3 sin 20001L"t + 4 cos(36001L"t + 60°)

a. (2 points) What is the fundamental frequency of x(t)?

[=E~~~~ CGNoM COMY\I\.JM.1>r \I~JtJr lJ+ IDOO ~ 1(00 a~ )

b. (6 points) Find all of the complex exponential Fourier series coefficients, in polar form(magnitude and angle), for this signal., / t\

jll'\,. e ~ CtJ5 LoB -1~o)

1- (of) ~ 3 Cos ~t - 9t)&) ,'- if c.~ (~~e!~'= .,I.- ~ 6'c)

5 i.() 0 q tJc.

[ L\ Joloor;'1rt -jQO8 -';'60D7rt '9"~ 13&Dotrt- j(qoo -j3fo()[)fft -jteO6A 9 ':;. (/5' e.. e +-- ItS- ~ J e./ 1- J. -e... e. f-~ t e.

=) {;(s = (,5 ~ tl.-s =-/, 5m 0 41:" 02&0 4:_q : oZ btJ°

A)l ~1f./.Q t ~ fl =-0~---

c. (4 points) In the space below, sketch the 2-sided magnitude spectrum of x(t) and the 2-sidedphase spectrum of x(t).

~ r---l ' f ' 1.<.. I, f;"' ,!,i, r 1.5""

~--+ ~--- -.- -

-{~OO -(000 0 100D f/5fJo "c(U~)

1 °0° I ltlG6

--- r--=~- I? 0 0 ~~~--t i 1~ a ~-L---/06 db. Ie-DC) t(~l:--)

-lo6~ 8

-90

Prof. Petr Spring 2005

EECS-360 Exam #1 - 4 - . Name: t F- i1"'0 =-lJ,l. to::.. 10 H~

r4. Consider a real periodic rectangular pulse train x(t) with period 0.1 sec. For this signal, the

complex exponential Fourier series coefficients are (for all integer values of n)an = 6 sinc(O. 4 n)e-jl.2nn

a. (2 points) What is the average (DC) Valu:~~~~:;]Of x f)?

. -'0~ 0':'" Co SI f\ c. (/) ) e. j ':- ~ ::. q bb. (3 points) For what values of n (if any) is an = O? .

f:O(,.t\~ ~r (L. suci ~ d,q,,:. I~~er/L - + r" 'i"- 10 .t /£ .. \. l ,- -~I -) }

c. (4 points) One term of the cosine-form Fourier series representation of x(t) isA cosG(;O}t + 8). What are the numerical values of A and 8? A must be a non - negativenumb"!;' and 8 must be expressed as an angle between -Jr and Jr.

, 3D 1ft- !4 ~ ~ ((.( 3 ~ e-::. &':;. ~{o / - 4 _2~..~i).L217I - J' It 2. Tr(;) ~

- C1 ~~--~~-~---~~~=~~~ {),q.3S"~rr ::..- fJ.q3;;-f~:..~ [[ --=;;\ t4 ~ (~) LD,q 3 s--j.:. I,?""

e.:. -{)t~7T

d. (6 points) A new signal y(t) is formed from x(t) as fOllows; y(t) = 2 x ~)- 8. Letfo represent the fundamental frequency of y(t) and bn represent the nth complex exponential

y

Fourier Series coefficient of y(t). Find the value of foy and an expression or expressions for

all of the bn values. r>'...,... :--. r. ~ I 0 J/")7~-;.I-lJ I rTi ~ '].l.,eg.tJ.JJ..M ~ , 0 ~ ~ loT 0 X '"=' """ i' 0 j:' ! 0 ;: {i; J-':~~ I

b /> =.. ~ q t1 - ~ ~ I ,( - J" .=..!1~--~~ 1- - - .

IvI (J. b - ") - J'~rrf'Lfl>l{lDI3\ '1/:1 . - J O,lon-1'L

N l 6' i'"N ; fl - 0(.. Co\' f\... t - J V -= o<,.£..{ VLe..

! ~-~ ~_..~_.~-_.~.~-_.:~. ( \ - J . If {> n- ~

t~=--_:~~~~~~-=---JProf. Petr Spring 2005

EECS-360 Exam #1 - 5 - Name: k E.. j

5. A real, periodic function x(t) is sketched below. In the range -0. 5 ~ t ~ 0.5, x(t) is given byx(t) = 3 cos(1Z"t). r J.1

7 0 =-'I to:"t>, 1 5 IT C0-. x(t)

~ +-{~~t + +~r:~+--+--+j)---j - I I I

I

--f-- --:-- -1---5 -4 -3 -2 -1 0 1 2 3 4 5 t

a. (4 points) Give the specific fonnu1a for the complex exponential Fourier Seriescoefficient a. for this specific signal. You must give specific numerical values for allconstants (ex~ept tr), but you should not attempt to evaluate the integral (for thispart). Nt>k: ;X (i-) '&..0 t!!9l ~ 4~ . r;;r j t>,.s-- - jJft

-:::= J g02 r To/2. qz.. = -;; .3 tt)'srrt: e.. ~t

q~ ~ -r; X [t-) ttYJ &zrf\. to 1:-) dt -0.5

0 61,S ( O(S

4~:. CJ,5 r 0 .3 ~tYJ l1t /:.) CD Sl7T~) d t -::. 1,6 J D LoS ~rrt-~dt-

b. (4 points) Find the numerical value of a" expressed in polar (that is, magnitude andangle) fonD. That is, evaluate the integfaI from part (a) and express it as a complexnumber in polar fonD. rlJ,5 I )t ":.. I ~ To:- I So J. :::.. 0 I ,- 1,5" r Ot5'" ~ 0

Clz. - T )0 [{ f C ().-":) ~lT& 1 dt

-:::.. {)(f5" (t91S--0) ::... {~~~~

Prof. Petr Spring 2005

EECS-360 Exam #1 - 6 - Name: t f. ~,

iw ::. f.r;O~ Jr ,"'" \

6. The CT sinusoid x(t) = 8 cos(6007Z't + 7Z'/4) is sampled with an ideal !C-D converter toproduce the DT signal x[n]. A portion of the 2-sided spectrum of the sampled signal x[n]is shown below. Only the range 01 DT frequencies aJ between -37Z' and 37Z' is shown.

: Magnitude

, - ~ - - - - - - - - -~- - ~ - - - - ~- T - -- - - - ~- _T- - - -,- - - -:'1- - -- - - ~-\ -2. 87Z' -1.27Z' -0. 87Z' O. 87Z' 1.27Z' \' 2. 87Z' aJ

~ : Phase

- ~ - - - - - - ~ - - ---:~ 6' 2~ - ~ - - - ~/~- T- - - - - ~~~~~ _T- - - - - - - - - - - - -~'68_7Z'- -2.87Z' -o.87Z' I 1.27Z' aJ

-7Z'/4 -+-I

a. (5 points) Give a possible value of the sampling frequency Is that was used tosample x(t). Note: there may be only one such frequency, or there may be many.

~o (p~IV\(l p~ liT FI'£J1J :; ~.prr) bit All..! tlfpoJt# pf'i/Joll. (krdrrJ ~ri~) -

:So ~ ~ I ( z'7T: "*.0:) rs.:. ~fJJ! ~[~~~~~~rOO 1ft=-- r 5It z.7I--~

j(t)b. (5 points) ~. value of Is tha~ you. ga~e ~n part (a), write an expression for the

output;;~~f ;~-Ideal reconstructIon CIrCUIt (Ideal D-C converter) that uses the samevalue of Is that the C-D converter used. . . 11

(/(QC~UL+ !o~.Jl.Q.d bl\.- fC1--~~f2~::!; tL {(lo :w ;::tJDf~.:. (0 r flT~ (504)'::: 4tJO rr

I~~~~;~~~~~~~}c. (5 points) Is your answer to part (a) the only possible value of Is that could produce

the spectrum shown by sampling x(t) given above? If your answer is yes, explainwhy. If your answer is no, give one other possible value of Is.

/I. 11_1 ./ f. A ~~E~~~a alJ-o fJ/( w ~ 1,2 rr +J.1T =- 3 t 2 rr =- --t5

=) +s:.. ~ :.. I!~~~~:~

~)~ Prof. Petr Spring 2005

c..C

c

EECS-360 Exam #1 - 7 - Name: t E :J--

7. A certain DT system (call it System X) is not necessarily causal or linear or time-invariant.

When the input to System X is o[n] the output is 30[n - 2] + o[n - 3].

When the input to System X is o[n - 1] the output is 30[n - 3] + o[n - 4].

When the input is 2o[n] - o[n - 1] the output is 6o[n - 2] - o[n - 3] - o[n - 4] + 2o[n - 5].

a. (2 points) Is System X causal? Your answer should be one of the following: Yes,+ {: No, Possibly You must justify (give the reason for) your answer.

~tP~ b p~: (1.tJM!o.(L~ is {~(~~d ~ tJ..(3 ;1tP«f-O(;(.{f~

.fi~ ~;IJ('~ fcrd, bvl .S'r)1.re fIJ-e.. ~o f\?)-f ~ ~-J~!iJ4~vk !J ! ~ t f JI) f J - tJoJf III re (a..,~ ..s f,.; p) wt r:..o/l Nt. 6!JJj ~ r StJ N( ,

\t\ ¥-l;. b. (2 points) Is System X time - invariant? Your answer should be one of thefollowing: Yes, No, Possibly You must justify (give the reason for) your answer.

p~; nM,iJ (~lc.o-~ ~ th .[rNf- oZ..

",-,~,- ~fl)} (b'( f"d (.sA,.r+ed I~pd rl\?JJf:! 'r f'\.

.6kl~~d Or.J-~)) hCl<c-f- UJltf\.o--t ~~ ~ r ::,tl ~ '

c. (2 points) Is System X linear? Your answer should be one of the following: Yes, No,Possibly You must justify (give the reason for) your answer.

!J2-: I F ~ IAJM _li_t~4.[) re.i p~AI1IPI- f ht5l [["J - 0-[1\-/ J Ct\)f>AJ1.d ~ ;( &d~"Z] + [[/1.-3] 1

- [3"J;[n-3] .f o[",-trJ]-

~~;J ~ (p O~-2-J - 0-["'-$] - f~-t.{]

bJ ()c.,.,,14A.J (Jdfd NJ t1Ndr i<LrM.

Prof. Petr Spring 2005

--" .", "_.~-'""--""

EECS-360 Exam #1 - 8 - Name:_.( £- t-

8. A DT finite-impulse-response (FIR) filter has impulse response h[n] with values of -4 forn = 0,3 for n = 1, -2 for n = 3; and 0 for all other n.

a. (2 points) In the space below, sketch h[n].

() t3 .3

b. (3 points) Expresss h[n] in terms ofDT impulse fundtions.

~ lfLl: -40["J f-3 6 [A-(J -,J cr[ft-37

c. (1 point) What is the support of h[n]?

R ~ l ~ f- fl.. +f' i}.ItIvt D -10 3d. (1 point) What is the length of h[n]?

L/e. (3 points) Write the difference equation for this particular FIR filter.

!j [it] = -1 ,>(["1 f-j x[i\-tJ -~ X Ln-3 Jf. (4 points) Find the frequency response H(ejaJ) for this particular FIR filter.

A ) ~ . ""'

'w ' - fLlAJ

f?{~J -::: ~ J\[tt] eJ~ =- -0() A 1\

c> - J" (,J - J.3 (..J

::.. -iJe. +3e. -).e,.5 ..(. .3 ~' J" bJ - . tJ - J W

ff-Le :::-- -~ +3 e.. J -) e.

Prof. Petr Spring 2005

'I , .

EECS-360 Exam #1 - 9 - Name:--K E ~~

g. (5 points) Use the tabular method to find the output y[n] of this filter when the inputis x[n] = 38[n + 1] + 28[n] - 8[n .2].-

(l ~ -2. -{ D ( ~ 3 'I ~ ~(p

t[r 1: 0 3 ~ 0 I {J V D 0

_-!:J!!:1j--~ E-- - '/ 3 D - z:._~-~~h[t> 1 XL r\ 1 ~ D - (L - ~ C? L/ 0 () () C

n[t1 x [t1-J1 ; DOg ~ D -3 0 () ()

h[31 ~[f\-3] ; 't) () 0 0 -~ -'I () ~ 0

, ~ L t\ 1.' t;> - l z.. I ~ - L - t D ~ ()

~["1: -11, f[n+i] toLnl +~t;[it.-'1-,,( o~-z.3 -1-~[ft-3].+Z;;;["-s-

h. (5 points) Draw the Direct-Form block diagram for this particular FIR filter below.

4x[ttl

~[fI. -11

)( U-ll-;l, ~ L"'~

¥ LVl-3

Prof. Petr Spring 2005

Pi

EECS-360 Exam #1 - 10 - Name: ~ £ 'I/)~Nl:d :J.7rT(

9. The frequency response of a particular DT FIR filter is given by H(ejW) =~-IJ:..; pA4J{ S/,p.ea. (8 points) In the space below, sketch (separately) the magnitude of this fr~ency :.-;.

response and the phase of this frequency response~ Show only the range of DTfrequencies between -7r and 7r. For the phase response, express all phases as values

[between -7r and 7r. I1\l A'"/M-e [solid 1;1\Q) (

'1

I-

ff ,,\ At::-lT - ".9-':"Jr/"2 0 7r!-'" ',' 1r I.J

A -;;11" -- r r ~IIAAA ( rl- ~,.,.. -~ - -- .-'- -~... i CD's c.,\Wr, eIJ ~l~ I\., ;.,..

5(1. b~~(, ~ rr n-CCO-G i5 1\~jtl~) /" 5ltp~ -:..-i.~---Tc~S c..'~e.o ~~~----10 1~+ 7r --t J , ~..~_C_-" , ,' -,.'t---

~~ ~ ::. -1r -rr slo e~-z. -7/'2.. c b1

-rrb. (5 points) Use the frequency response to find the output y[n] of this filter when the

input is the DT cosine, x[n] = 5 cos(n7r/4 + 30°). . (~, \ - "rr/z. /.1[ ~ 0\~ =:-1r /1 H (eJ 7T/4) =. C /J? LTr/4) e.- J J. 'f):. f) e '1-01- e.J l Z - CJl> )

~ [1\ 1 ~ I t{. (~ j Tr/'/) (£ c. 0 S' ~ A. ~ +-';0 0 +- ,~~__tf:.(~:~-~.:~~:) J

[j~~~~~~~§~!~~2}c. (7 points) Find the impulse response h[n] for this system and express it in terms of

DT impulse functions. . "-It (e. J' 6:J ):. ~ ~ r t\ J ~ J ~ IAj r n::.. \ ( /1.:".3

f\:..-~ A;'"

ff{ej"()j.::- lSIt5lejW+e.-J'W) ~'.1~= ()t5e.-J'l...)te,£-E>--j.3W

ry hlI1::0,£) ~L3]'::O'~J oAJ d~r hLJ\]'::[)

1 h[/l] ~ CJ,56[n-IJ + ~/S- J[A-3]Prof. Petr ~"":"-': c--"---~ Spring 2005