This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **Problem Q2.1: Consider a periodic signal with a fundamental period of 10 seconds deﬁned over one period by O, for —5 < t < —3
:1:(t) =
It], for —3 g t g 5 (a) (5 pts) Draw a labeled sketch of a:(t) for —10 S t S 10. do ’5' ’3
(b) (8 pts) Write the Fourier integral expression for the coefﬁcients ak of this speciﬁc signal :r(t).
Do not include any absolute value expressions in your answer; to do so, you will need to
express your answer as a sum of two integrals. Set up all the speciﬁcs of the integrals (e.g.,
the limits of integration and the integrands), but do not evaluate the integrals. (Because
of limited time on the quiz, we are just testing your ability to set up the needed integrals, not your ability to do the calculus. .LLO l‘ 1... Ans-L82 XLALA’ OM- “) v
‘5 .
435:“ y a?” \ ell rﬁlr =L.a+§§=_ﬂ.
'—' l5 1‘ 2.0 [O 2' 2' “°\. ((1) (5 pts) Suppose we create a bandlimited approximation of m(t) by summing its Fourier series
for a ﬁnite number of terms: $BL(t)= 2 die ej(27r/10)kt
k:— N If this mBL(t) is sampled at a rate of 10 samples per second, what is the largest value of N
that could be used that would not result in any aliasing? “N [V I l' i :: ..__.. .1 11/.
H1“? We“); ’\ XgLi) 5 210 S to H% W ”Wk {at 27” Z‘ “i”; 4J0 wﬂl 1071'; Problem 02.2: Consider the following cascade of an ideal continuous-to-discrete converter with an ideal discrete-
to-continuous converter. Part (a) is independent of the other two parts. X (t) Ideal Ideal
— C—to-D D—to—C
Converter Converter y(t) =1/fs Ts=1/fs (a) (15 pts) Suppose the sampling rate is f51 = 1000 Hz, and suppose the output is y(t) =
cos(27r(200)t — 7r/ 5). Give three possible input signals of the form a:(t) = cos(27rft+ (15), with
100 < f < 1500 and -—7r < 45 S 7r, that might have resulted in that output. Write your three f, ¢ pairs in the spaces below. (Hint: one of the inputs results 1n no aliasing, one results 1n aliasing without folding, and one results 1n aliasing With folding. 3C) 2“,“[5- - IT’S-'3 ”wk va~1= f = mm, = _____ looo “5(u.“ __n—(r) ____._____...__., wast??? «”158: 950% Jm'Tr/SB For parts (b) and (c), suppose that the input — cos(27r(400)t + 37r/ 7) (b) (5 pts) Specify a sampling rate f3, With 250 < f3 < 600, that would result in an output signal of y(t) = cos(27r(100)t + 37r/7). NL¢J\
Zn" ’100 + Zak: 19.9- 1" ‘Q’ 5"“
fs=_;3_-__o._9___ samples per second '( {3 K
S =l00 =7 H00 + K {A
at,“ {5.300164 (0) (5 pts) Specify a sampling rate f3, with 250 < f5 < 600, that would result 1n an output signal ofy(t)=cos(27r(100)t——37r/7). Now We, W
”373.0,? +2.tr\< ’3 ___0___(0 Zr 5'00 f3 = ___________ samples per second 45 4S
a: ‘too +K {a ""0 Problem 02.3: For parts (a), (b) and (e)1 consider a discrete-time system whose output y[n], given input :tln],
is speciﬁed by yln] = (A + $[n + B]) 803(Cn). In each of the following three parts, give the least restrictive description; i.e. if the system has
a particular property for a range of values, be sure to specify the whole range. This may involve writing things like “D = 3. E > 4, and all F ,” except in you’re case you’ll write Something about
A, B, and C in each part. (a) (5 pts) For what values of A, B, and C is this system linear?
Let $th = olxlh] 4— [5&3] I he“ 58"] ._ (A+ a. Kl.“ B] + [him BDcoSCCnl
In ”this 555% XI“) |——7 Sin] = (A+ it‘ll“ 3]) ”5(an and
Mn] H 3112“] = (M THEMEJ) COSCCH) . 5cm a gene” s stem
must SOiﬂrl-‘y xEn] l——) OWEN] 4.- {391B} , “linen A“ (b) (5 pts) For what values of A, B, and C is this system time—invariant?
A Sgstm l5 timeﬂnvmimit [£- xEn-na H 51“- “a .
I“ this sgs’cem xvi—nu] H (M xtn-m+ 8]) (Le-50:“) , (e) (5 pts) For what values of A, B, and (7' is this sysaem causa‘? A SgS-tﬂm I3 cause? i-F a Sgsfmk MPH} doe; ng-l: dermal an 9W9, in-ud. _ _
Thus B 50 , A it. C cur-ﬂ no“: rﬁs-trr’ctecl- For parts ((1) and (e), consider a discrete-time system whose output yln], given input gel-n]! is
speciﬁed. by -y['r1] = 33:]1t] — 21-[11 — 1]. Give your answers in terms of delta functions. ((1) (5 pts) Find the output y[n] if the input is :1:[n] = 45hr] — :il'n. — 1].
Since 3 [in] = 3 x [n] - 131314]!
{km Mn]: 3 at] _ 285mg, m§orm cow we“
13%]: 'XL'n] *- hfn] / hin]=3s[n]— 7.51m] , 5.2““; the, length or the meme F€$PQM€ [s ”the, length of the exhibit; the {“(Ni— ‘I5 0x
6'1an mime. 9cm: Ll5[n+3‘l] {n Mu to Producﬁ “the j§u€n 1:]th Problem Q2.4: Match the time-domain system descriptions in (a) through (f) with frequent}r responses (I)
through (V) by writing the correct capital letter on the “Answer” lines. 3 points for a right answer"
0 for a. wrong one (i..e. you will not be penalized for guessing.) Hint: If you ﬁnd yg'aurseli‘ wanting to select “none of the above” more then once‘ you might
want to try starting with the frequency responses and working backwards. 3m Lemma Pam) (Dx‘r i chie 't Form) éﬂla-E. runner-v13 (n) Mn] 2 n[u] — ruin. — 6] Answer: ___S____ (1)) Mn] 2 riﬂﬁhi] + 6[n — 1]) Answer: --m_ E‘Pi‘, ON efajﬂr
. a- H3
.- . . w - J
(c) Moi] = 5m] + (ﬁn + 1] Answer: ___LA_ 83) deﬁm‘i’te“ [fuel ) - e a + L
.5 :13 .. -;W( J“- 3’3””)
_ HLeJ‘” = 1-91 , 9 .5? A
[(1) Mn] 2 ﬁle] — of” — 2] Answer: ”hi ) : 2") LE’JW) 5 .2“ E w) . Hal's) :: [ﬁe-l; ([404: available.)
51,4; running Swim! but no biﬁf—l’tiﬁl germ
. Answer: ___B____ amoebic. h . n .3».
. I ‘63 .- -jw :1qu *J
'Htel ) — [+6 +e Answer: L (e) Mo] = 6[-n.] — 5hr. — l] (f) Mn] 2 {tin} — nin— 5] . si11(7£:'/2) 2-; _ '
Wexpf-—de) (II) HEW") : {K} H(ei¢) = 1 — BXIJULD) (L) H(5J“") = meme/2) zéiﬁm antennae) (M) Heir) =
(P) H(ej“;’) = (305031) exp[~jt£r) (R) H(eﬁ’) = [1+ 2cos[:£=) + QCUSQQH exp(—j2£2) )exp[—jLIJ/2) (N) HEN") = 9jsi11[‘u3)exp(—jtb) (Q) H (air) = 2 (men) e-x;>(—2_-;e) Sil‘1(3t;‘) sin(JJ/2) _ w r (8) Her) = mam—mm (U) Iﬂeﬁb) = 1 + eprLEJ) (V) None of the above Parts (3;) and [11) may have more than one correct answer; but you only need to find one. (W) [—1 pts) Given the frequency responses (I) through (U) above, list One system that would D result in un output of Mn] 2 {J for an input of .1r[-n.] 2 4'2. Since we] is a we had (“new Fretsuence) 471:) 5550
Page time: (h) [3 pts) Given the frequency I'm-313011585 (I) through (U) above: list one system that would
result in an output of ju[-n] = 0 for an input of :1.‘ [n] = 95111£ﬁn — Tir/Q). 11“] NJLS A in] {hide 535W!“ 0th L5=Tr_ in e N)“ Ejsiewx. our] f’tgsihle QMNQX‘S: ‘3! MIN? 5 1M] ...

View
Full Document