maths

+. e, Let Z-

,z t- ,- -t,' 2 7.;,l-'rr a "., ':,

o-2 i.11 '-. . 'z' '.7 '. '+ >.zo

Cont nuous-Time Signals and systems Chap 2

\-r ' . D(Jl'.\f "'i|u l' \l '' 'i-r'l' 'n1l|''r-'.,". , .;- ,;;" . for thesc oeLiorlic signals is Prcsented' -i' ",. '' ','. " ^ l "

..,r'"" " u-r, rr or 'r curr' n!^u' ; ,1-;.",,. '-' ,.'. ' -. q-"i ,.o'.' i\' " "or'IJ-

;' . "i " " ' u' 'r'' 'cJ orucc'lu

i,,. $ h-.n<r .

" ., r.,-,1t,." ll' ',;"' " 'rr r'' J +''qrr' t r C'"otLr '-

".,.:,t" ,. '.] r 'c 'e rrr'< 'e ' J "'t<"

= i,r"niicoI t no"icr in'ome cases thcre arc signiticanl difici-

)

PRORLf}'IS

Section 2.1

2.r: Ihc liqnrls nr Frgure P2 1 xre 7'ro o\c'Pl 1i <h'\rn

'rr) ,( ,t:tt (ii) r(lr 6 (iii) r(l + r) (ir) ril ') Veril) yotr! rc\ults hl cneckin! 'rt l'rd 1\o toiits

Chap.2 Problems 79

rchniquc i:.

n Chapter -

licant diff.:

(b) Repeal PaLi (a) (c) Repeai Part (a)

for tlre sisnd r(i) otFigu.e P2.1(b). for the sisDalr(r) ot Figu.e P2.1(c).

2.3.

?.2) The sismls in Fisure P2-l arc zero except as shorn.

(a) For the signal j ( 4 of Figure P2.1(a). plol (i) 4r(r) 2 (ii) zxlt) + 2 \iii) 2x(.2t) + 2 (iY) 4x(r) + 2 Vcily yourresults b) checking at least tto points

-{1r,) n"p""t r"'r 1o1 ror the signal r(r) of Figure P2.1(b). G) Repeat Part (a) for the sisral r(r) olFigure P2.1(c).

Given thc two signals in Figurc P2.3,

9.! E\preqr ',(r) Js a iun!r,n otIr1.1.

i(b))Vcily rour resr t by checking at least lhce points i. tinc.

ii.,t, civen thc sienals r(r) ard-r'(r) jnFigureP2.4, (r) Express ! (r) as a function of r(r). (b) veriti )our results by chccking a! lea$i three poinls in time (;i Expresi(4 as a lunction of](,). (d) Verifythe results oi part (c) bychcckingat least threc poinls in tine

Section 2.2

i.sl Ploi thc cven and odd palls ot the lignal of 1('i Fieure P2.1(a) ,lbinig-. P2.r1l;

l\I\ \t\

I I

80

5

2

I

I

a

8

2

0

10

Conlinuous-TLme S qnaLs and Sysiems Chap

0 a 21 LO 1:

---,.-t Ll

l0 a0510 {h)

15

(c)iFieLrre P2l(.) (ii).;Figule P2.a(ur iery.rilv )oxr.csull\ Ning (2 l1)

!.6. For cach ol ihc signrh lircn, detcrnrii. nialhcrnaticrul il lhe signal is crcn odd nelthcr Sketch Lhe \rirciorns to vcrify 1_our re\ults

'i"l .t,t = q, ib) r(r) =. '

Chap. 2 Problems

: sln(3, + ri) : a(t) u( t) : ult 1)+ul L t)

/ 2.7.)The alerase valuc ,1, of a sisnat r(/) is aiven br

Llti,(r) be the eveD part andtrD(r) be lne od.l!a ofr(r) (a),lshow that

io.'(, Cti.ta ,9't'l (r),r(r)

(b) Showthal

litu

(c) Show thatr.(0) : 0 and r,(0)

|!.-+ l',

;r":-!, l',,,,r,ta, = u

* l'7u," -4, :

'(o). ,.29 Gne prooti of ftc rollowing sratemenrs:

ia) rte sun or Livo even runcrions is evcD. (b) Thc sum ol two oddlutrcrions is odd. (c) lhe sun1 of an evcn luncrion and an odd tunctioD is neirher cven nor odd. (d) The producl oftFo even lLrncrions is e!en. (e) T'e p od, .1 u ,\^ ^ d tur. rr. ,. i. p\. r(f) Thc producr of rn eleD lxncrioD dd ar odd tunctiotr is odd.

2.9. Cire! in Figue?2 9 are the parts of a sjgnat r(r) and ils ocld parl r,,(r). fbr / > 0 orly: that is, r(r) and r,,(r) for I < 0 are nor gilen. CompteLc rhe pror oi r(r) an.j a,,(ri,:rrd'( rdu. rl-e.er o c,.. ^.,t,,..,*,."o r" o"i,,gcach part ol the sianals.

l1lj,.?rcre ilathmalielly tha! rhc sisnats sjaen arc perio.lic. For each sjsDar. find l1lc fun_ damentrl pedod 4 and thc fundanentalfrequuxcy o,l

(b) r(r) = sin(& r r0.) (c) r(i) = err @Y'14 = ".s1211 + sr"1:,1(e.) r(t : at(i0r+"ir) (f) r(4 : d rln .r']ir