11.301 The pUIP'J5e of this problem is to numerically verify the sampling theorem relX",,,trnction formula in (1.4.5)_ Co""ider the following bandlimited periodic signal which can be thought of all a truncated Fourier ""ri""_

x.(t) _ l-Zsin(1rt)+cos(21rt)-t-3cos(31rt)

Wrjl,~" MATLAB ",;dp, which 11,es lhe futle,ion ·lLsine from problem 1.29 to approximately

l'ecoru;,ruet x"(t) a.o follows,

" ",,(I) = L "'a(.l,'I'),ill"["'!,(I,- kT)] .~-p

Doe a sampling rate of !. = 6 Hz. Plot "'"(t) and ",,(t} on the "arne graph u6ing WI points equally spaced over the interval [-2, 21. Prompt for the number p and do the following three cases.

(a) p=5

(b)p=lO

(e)p=20;

Solution

:t. Problem 1,30

;: lnitialiu

d' elear '.a ~ in line (> I -::!*atn (pt.t) +CM (2.pi .,,) +3.cos (3'pl''') , . ' t ' ) ; 1" ~ 6; T ~ I/ta;

1. R.econstrun "_,,Ct) from BEmlplM

p ~ f_p

~UII-;

(,d~~,',u-x,) puall"T (, (~)",', (~U) ~,'UO"l~u,,~) s1BQ"CJ

!(d',P1,. d gg."n uUHon.:qglIOOB\l 1'In.nd,) HUpd" ~ ""Hrn,O (0 . t' ,lI1PJ!\&lIJ1, 'd-,,':l. ' (:l.) ,,-,,' ~) 101

.. ,0 0·'_ '..' ( •

(\ (\ •

\/ \; r,.,

'. J• ,"->< • 0. ,•• 0 ";;

r~--'-~.

r

I ,

\/'(\

!\/

/

.'

•

•

.'

0·'_ .'"

•

( • !

I •

•

0

0

[!;il \ The Huu"".mrth filter is optimFtI in the sense that, lor" ~i,'~n filter order, the ITl''I'";lllde ~ ,""spon"p. i" "8l1at ... po",dbk in I,h~ """"band. If ripple; me "llowoo in the pll.'lsbllnd, I,hen a.n analoJ; fill,~l- with a. sharper cut,,,IT '''''I be achieved Con"i, ]", I,L~ fol1,,~'inM ('h.by"h,,,· J flit er thaI will be disclI ..ed in dctnil ill Chapter 8,

12837 Ha{s) s" + 6.1s' + 87.8.;'" + 2ri l.~s" + 934 3s----:;-T2lj:l.7

Wri,e a MATLAB script the use, the FDSP to"ll.H'x fuucticn f-fre~.' to c"ml''' l.~ Lh~ m..gnit ude m't''''''"' of this filter, 1'1,,1, ii, ov"r the range [O,:J] H", Thi" filter i' cptimnl ill I,h~ oen.e that

I.lJe paBl5ba.nd. rlppl"" are lill of the "arne si""

Solution

7. Probln 1.31

i. lnitiali:z;e

do daar N ~100; ~ !",a~ _ 3,

b - 116~.7 a. [I f'./ A1.8 1"1.~ 9:;.1.3 1~53,TJ

~ C~pnte ~nd pl~~ masn"ud. r4ep~n~c

["_a,f] ~ I_! ....qs (l>.&,N,fmHl;

"_a ~ ~bB(H_a), flgu'r"A

pl"t (f ,~_~) Lht.aa ('Mognl""

"

~a~l!d I-MQ.IiAq"1{3

, " , IZI' \, ,

" \ ~ \, ., \,

" '.... ,_...0"' ____ n

{o f ~ Iz} OOH

Z-"~+"Z'J &z~

~JZ-t"~+F-"'I-~ = (Z)x

UO!lnIOS

[(z)X JO aJU,,'il.l"hlL"'.I" uOlj!aJ 3"1"1 l""M (G)

·z LlL "l"Jllmll"l~d 0"'1 10 On~l " ... 11 ,"",,!>;, flu" '( z ) X 1UJ01SU",Jl-Z 3'11 P'",{ (,,)

t" + ~(q)'Ul "1: -I" zl(q).oov z]

z(q)Ul'"

__ "D +.(q)'roVZ -~" Z{[(q)SOOIl z](O)Ul" 1 (q)Ujs(e)'oov}

~"I z(q)""'''z-"z 1:,,+z(q)"""VZ-~' z[(q)'rou ;](O)"'~ + z{q)U19"(e)'oo '"

(('yq)00' .v1z{/i)Ul" + {( ~q)Ul' ~,,} z(oJr.,,, '" {[(Ii)"!" ('i') )"O~ 1-(Ii)"00 ('fq)U!"]~v }z

zlJi = (J U'"I''' l'1:'Z '!q'''.L J" "1(',1 A.l1ua 01 "~l\paJ (z) X l"'n .lJ 'J~A (,) '0 0 nall'" n:1: ~lq".L jO 3.J'I1 "'lU' 01 \l

1-r< 1-<--I

tll 'll (1+%)(1-%) -

,-=

, { _ ,.% = {; (·)x

""""1 ~,.. 8II01p"Jj 1"'l",,1 "'ITJI (z) X ~u,pu1ldx", (q)

L_'l o-"l ,_"7, ,7

= (o)X

61 (,) X .I" LllJOJ paJ01~~J ~'lJ,

o T-,,~-,

_.' WIt

" ·r_' (')X ill!]

..

(O)x

=

'-=II-',~ ,

I--'!(Z)x!, +0) ;

1=ll+,,

L-'I(i)xtl 'J

(o)X

~

~

~

'" ~

W

, ,U )

~

lb' ~

'H

, ~t_ ,

l-~[

I-l " r='1 (+ r

I\ -~,

[c-'ll_l~(7)X(r -")

o ~-, (Z)x ill!! (0)"

(f+Z)(f--=l = (zlx,

= (z)x

IaWl]" ["a' " .... "'.,,"u» 1~1J~ moil "'ClJdX',;[ (If);( puy 0. i':·t·~ '111'1' asa "",.'O],,,,Ul-Z ';lul",anOJ ~'11 JapT~uo::> Ill!]

(1 -~ ),,(~/ ~'1)sc

(,-"o[ ,-] ~ ,JJ

I~.141 Conoider the following Z.traruform, Filld ,,(Ie). ~

X(,,) = 5~~

I~I > I(z' - z + .25)(7. I 1)

Solution

The factored form of Xl') i"

~z"X(,,) =

(z - ii.WetII)

Using the r.,,;idne method, lhe initi,,1 value "I T(k) '"

:r(0) lin' ..'de)'_e< ,

1'1" n""lt,~. 01 X(,)}' , ~! lhe l~'Q rQr.~ ~w ~

H, = ..i{(._f1.e,),x(,),~ ')Iro.\J, d r,~,""11'

,10 t ;+:1[1,=0, (t, 1)5(.· I 21'.~:' _~zH~ I

{

'" -(-1--:-S1"

l ~c:) (_11 k+

m

('f)n\t+'r(I-)z+l',('ro)(~+'¥\:)1 (~T)

{l ,¥)n[,.+.(I-)Z+l+l(gO)(g+~E)1 (aliI) +{'l')~g = (I - '1)n('H + 'H) + ('1)~(o)'" ~ (~).",

~ C",n";d~r the nlilllin,o; a""m~e ~l',~( ~r ordcr m rrm" S,di"" ~.1.:\. ~

", 1I(~) -'.. Lx(k -i)

'" I] ;-G

in) Find (hi' ""n,r", (,U","""" Hlz), EX]J"'''; j,"" a ratio of two Pol,Yllomi,.ls ill z. (b) U"" th~ ~""m~trk ""ri"" in (2,2.4) t.o show tbat an altcm"tivc form of t,he 1,""'"1'",

[""etio" is as [DlIDw". Hi",· ~:"pr,-,,,," l1(k) 'c, ". rLy

~ m

Y (,J ",-'Xic) ~ ,~O m +'

Th"" the trausfer functi~n i.

y(, ~ II(z)

X(0)

• 'm

m+1L,-.

z-;

1 f-0 -1 f-I ,-'"•

m+l z_+_m_J ~ +.,,+]

• m+ 1

(1'1 ;;;"""""1'-",jl.iL I.h" iLi,,1

,m

'U(k) = m+l)L"'(.l' ;} ~"

, [oo ~ 00m+Ti t:",(k-i}-L ~(1'-'i)] ~ '-"'+1

72

I +'" (\ ill "),,, ('1)0-. +(\ ,,)Ii -= (¥)fl,

;(",c+,"",,""~!f[~,:~~J -T] = (,_Z-1)('[+W)

~h1"'1 a!," Z JO S.

- -- -

~ Consider a disc.rete-time system :{k' an example of h•.rmonlc forclnp;? 'Why or ,,'hy nol:'

Solution

(0) By i"-,,pe.el.jon, the. trankr luncl·ion i"

?,-l_ 1.6,-1 H(~l =

1_ ,_L +O.~4

(~'O -')(9'O-z)

"'S'O -, (t'O - ,)(g'O - ,) , (~'O .)~

(')X(z)IJ (,)..;:

(1 - ,/)n[1 + ~(t'O)\' - ~(~ro)~1 UJ (1 - 'l')n(~1i + ~H + TH) = (,/)"

c • O~

(~'O)(,,·O)

(~'O)O1:

'~'I(~'o -z)(g'O .)I ~'(S'o - ')01: •

L='II--'f,(z)";:(l _ z) 'M

•it·O) (~(,) (lrll )(1:'0 )

~(tO)(~'O-)O"

~'O---"I (I - -)(90 - zl • ~z(g'O - ')0

-

R, (z _ 1J.Ii)Y( "l,k-I I="~

27.l

I= (~ 0.'1) ,-,u;

2(O.rol l =

0.' 10(0,6)<

R1 .. (0 (I.4)Y(zjz

"atUn lj11"'" .",on-1"'1'1 l"~'~!II"0' 1?!,uou~l(>'''lllu\;l.InJ ;)tUOtUl'1>lj JQ ~ldU4"" Utl n "!'Il ''''A

(I 'i)"[~(O"O)OI -~(t"O)(JfOZ -Ojll =

(1-~)"{"H + 'H) = ('1)~

,ltQ](.ylJ'L (II) =

~(to)lo~I;I'll~-[l-+~lOl-J =

"

6!.

(z)YI(2)CH + (~)1HI ((z)Y(Z)'H + (z)y(z)1H (')A

uopn[os

'(2)~H 1m" (z)TH .I" "'][P""~~LJ(J;] PI1WU~ " p~[]a~ "! "jllL -l-ll'l ~mll!i1 ttl ljdaJll ."011 ]"lIll!" ~ljl ,10 lIO!l"lInJ J~J"UaJl n"J~1lO ~ljl PU!i1 Ijll_~ I

--

[I}2] Comidcr a di,eret" I.im~ "l',I.~jjj dfficrlbed by the following difference "'l\l~tion

y(k)=O.6y(k-l) I 0_1611(0, 2) + 1I1:"(k-1)+;,,,(k-2)

(a) Find the lran,Ii"r I'""d,;,m H(,).

(b) Find the impulse r""'pons" II(Jr:).

(e) Sketch the oir;rlFJ.l H"w ~r"l'h.

Solution

(a) From inspl.-'d;m,

10",·-\ +Sz l H(z) '" 1 _ O.TC"l hili) = 1,1 1'1", C",icl",,, "f H(z)z'-l "'''

R, = (, _ 0 ~)f{(")"l-ll.-o,"

10("+O~J,l-11

z + 0.3 ,_O.B 13(06)

, ,-,

m

,,

91'0

""

"

t_,(L'O .)( I -'Ilr +I-,(L'0-)~S -,(1:0-) (I +·l)t '" ·'11,._.'(1 'Y)r +l-"~ll-,Z(l +'1')1'1

J:"-~' I{,_,z(£ + zg -_q.)) "I' .• p

n-='Il _Z(Z)f{{L"1J rZllz~ = Iy I .', l'

~ko I ,) C+ zg ;:,.t<

6tO+2"t"! +,z (Z)HE+"H-,zt'

~! (2)H JO lilJOJ paJ01;)llJ ~'1.L (q)

,_76,,'0 + l-'VI + l t-'£ I l_z8 ~

,

,

fWfI

,-" Pl

,-, ,

" ·U011..mb~ ~~ual~lJ!p ~I\'I I"".~ I~)

"('Y)'l ,,"nod"".I ""I"Llw! ~ql pu1.i (q)

"(olH 1IL1!punj laJ5Ull/l ~'11 PU!iI ('ll)

= (~J!i

(t -'Onl. i(t·0-)(I -,y)£ + l--

"_z~,,'O -I ,_z"l + r.

';,,'0 ;:z (t+;:z)£

(I -'1)nl, ~(~'II-I-,_i~oll~nl + {·~)~f (1 -'1)"i'li + Ill) + (~'y(O)" = (,,)1/

[ ~(", H-)~L ~l -

"n--'I ','[1_0 =

T-,,"(t + ~zJ~ ~'O ='ll_""(2)ff(~O+Z) ~-~U

"

;. , , ,

g~-O-

" , , ,

(~-o -z)(~-o + z)(8-0 -z) ([+~)~~9

"1 (z)H )0 lliJOJ P"JOlO"J ~q.l

(s-o -_z~-O + ,")(8'0 -z) (z)H(l+z),zg

12.~71 Use (he J~ry t~",I. '0 find e. r..nge ofvnlllffi for lhe pararnete.r" nvcr whieh the [ollowingsy",em .~ is DIDO stabk. h .your answer oon,i,(eLlL with Figure 2.~,~?

JJ(z) '" " (, ,~T(Z2 ~ +0)

Solutiun

Sine" the pole "I, Z = 0.8 is stable, it i" ,,,rIicient to examine 1.11" 'ju.

.H ·(llIO PlI~ J! ~rp"I""! lIIa,""s aql '11'!"'tOllO" U< Oq ~,,~ lI~'ll ala:",s "'''''''"1 "'41 ~~"jS -I'> U > 0 '"""cL'[ 1"1 'j'll'!l ~'um,,~;>J pu~ !1(lP!S '!tOll J" 'I'"ll ~J'"nbs llu-!,\"~ ',,(1 -XI) < !(!U-I) 11 ,(j11.., ["''' J' ~"mood SjlOl'llJalllnU 8'jcL -~'''l''Q(j Sj JOjSUjlllOUap a"1 '1 ."> 1"1 CKIUjS

,'"-1

l,( I oj t(t"'-I)

;:u-1 I-V I~"I-I1_ 0 ,0-1 -"