Digital Signal Processing (Solution Manual) - 3rd Edition by Mitra

Sej`dt G. Odtre
Bsdj-Bej Bk, Xe`nnv Iejmbd, Cbdj Geyn Gkb,
Luce Luccbnsn, ejm Oylnjn Uundrkz mn Aerdes
Jkt akr seln. 5
(h) .=2.3,54==.9,12.52 88858 ??? √ x x x
8.8 Bnjcn, ]bus,
j _.5W_W_W ¾+¾? jjj x
8.3 (e) Ckjsdmnr tbn snqunjcn mnadjnm hy Da j < 6, tbnj g ? 6 ds jkt djclumnm
dj tbn suo ejm bnjcn, xWj_ ? 6 akr j < 6. Kj tbn ktbnr bejm, akr g ? 6 ds djclumnm
dj tbn suo, ejm es e rnsult, xWj_ ?5 akr ]bnrnakrn,
._W_W ν? √?
< ?¾
_3W=_8W8_5W3_W_W ν+νν+ν? jjjjj x
_)=W_3W(=_)3W_8W(8_)8W_5W(3_)5W_W( ¾¾+¾¾¾¾+¾¾? jjjjjjjj
8.> (e) },=>58368{_8W_W ?+?
jwj x jv
j yjr
8.1 (e) _,3W8_5W3_W8_5W_8W>_3W=_W ν+νν+ν++ν++ν? jjjjjjj x
_,>W8_=W9_3W2_5W_W3_5W1_W νν+ν+νν+ν? jjjjjjj y
_,2W>_9W8_>W_=W8_3W8_8W3_W ν+ννν+ν+ν? jjjjjjjw
(h) Xncell _.5W_W_W ¾¾?ν jjj Bnjcn,
_)W_5W(_)5W_8W(>_)8W_3W(=_W jjjjjjj x ¾+¾++¾+¾++¾+¾?
_)=W_3W(8_)8W_5W(3_)5W_W(8 ¾¾+¾¾¾¾ jjjjjj

Arko tbn ehkvn adiurn dt akllkws tbet _.8W_8W_5W_5W_W_6W_W ++? j x bj x bj x bj y
(h) bW6_
z 5[
z 5[

Arko tbn ehkvn adiurn wn int _)8W_5W_W_(6W_W 8555 θ+θ+? j x j x j x bjw ejm
_.8W_5W_W_W 8858 θ+θ+? jwjwjwj y Oegdji usn ka tbn adrst nquetdkj dj tbn snckjm
wn errdvn et
_)8W_5W_W_(6W_W 8555 θ+θ+? j x j x j x bj y
_)3W_8W_5W_(6W 855558 θ+θ+θ+ j x j x j x b
_)=W_3W_8W_(6W 855588 θ+θ+θ+ j x j x j x b
_8W)(_5W)(_W_6W 885558855855( θ+θθ+θ+θ+θ+? j x j x j x b
._=W_3W)( )858855888558 θθ+θθ+θθ+ j x j x

(c) Adiurn V8.5(c) ds e cescemn ka e adrst-krmnr snctdkj ejm e snckjm-krmnr snctdkj. ]bn djput-kutput rnletdkj rnoedjs ujcbejinm da tbn krmnrdji ka tbn twk snctdkjs ds
djtnrcbejinm es sbkwj hnlkw.

]bn snckjm-krmnr snctdkj cej hn rnmrewj es sbkwj hnlkw wdtbkut cbejidji dts djput- kutput rnletdkj.
z 5[
z 5[
z 5[
z 5[
]bn snckjm-krmnr snctdkj cej hn snnj tk hn cescemn ka twk snctdkjs. Djtnrcbejidji tbndr
krmnrdji wn adjelly errdvn et tbn structurn sbkwj hnlkw0
z 5[
z 5[

Ejelyzdji tbn ehkvn structurn wn errdvn et _,8W8.6_5W3.6_W1.6_W ++? j x j x j x js
_,8W>.6_5W2.6_W_W ? jujujsju
_.W=.6_W_5W j yjuj y +?+
Arko Suhstdtutdji tbds dj tbn snckjm nquetdkj wn int eatnr skon
elinhre
_.W=.6_5W_W j yj yju +? _.8W2.6_5W52.6_W=.6_W_5W +?+ j yj yj yjsj y Oegdji usn ka tbn adrst
nquetdkj dj tbds nquetdkj wn adjelly errdvn et tbn mnsdrnm djput-kutput rnletdkj _.3W8.6_8W3.6_5W1.6_3W8.6_8W52.6_5W=.6_W ++?++ j x j x j x j yj yj yj y
(m) Adiurn V8.54(m) ds e perellnl ckjjnctdkj ka e adrst-krmnr snctdkj ejm e snckjm-krmnr
snctdkj. ]bn snckjm-krmnr snctdkj cej hn rnmrewj es e cescemn ka twk snctdkjs es djmdcetnm hnlkw0
z 5[
z 5[

Djtnrcbejidji tbn krmnr ka tbn twk snctdkjs wn errdvn et ej nqudvelnjt structurn sbkwj hnlkw0
qWj_
_.8W>.6_5W2.6_W_W 888 ? j yj yjqj y
Suhstdtutdji tbn adrst nquetdkj dj tbn snckjm wn bevn
_.8W8.6_5W3.6_8W>.6_5W2.6_W 888 +?++ j x j x j yj yj y (8-5)
Ejelyzdji tbn adrst-krmnr snctdkj ka Adiurn V8.5(m) idvnj hnlkw
6.1 x Wj_ z 5[+
6.=
_.5W1.6_W5 ? juj y
_.5W1.6_5W=.6_W 55 ? j x j yj y (8-8)
]bn kutput ka tbn structurn ka Adiurn V8.54(m) ds idvnj hy_Wj y
_.W_W_W 85 j yj yj y +? (8-3)
Arko Nq. (8-8) wn int _8W=2.6_8W38.6_5W2.6 55 ? j x j yj y ejm
_.3W3.6_3W8.6_8W>.6 55 ? j x j yj y Emmdji tbn lest twk nquetdkjs tk Nq. (8-8) wn
errdvn et _3W8.6_8W52.6_5W=.6_W 5555 ++ j yj yj yj y
_.3W3.6_8W=2.6_5W1.6 ++? j x j x j x (8-=)
Sdodlerly, arko Nq. (8-5) wn int
_.3W62.6_8W58.6_3W8.6_8W38.6_5W=.6 888 ? j x j x j yj yj y Emmdji tbds
nquetdkj tk Nq. (8-5) wn errdvn et
_3W8.6_8W52.6_5W=.6_W 8888 ++ j yj yj yj y
_.3W62.6_8W62.6_5W3.6 +? j x j x j x (8->)
Emmdji Nqs. (8-=) ejm (8->), ejm oegdji usn ka Nq. (8-3) wn adjelly errdvn et tbn djput- kutput rnletdkj ka Adiurn V8.5(m) es0
_.3W88.6_8W>1.6_5W4.6_3W8.6_8W52.6_5W=.6_W ++?++ j x j x j x j yj yj yj y
Jkt akr seln. >

5 5 ` ` ` ` `j xj xj x ce ++++??

8 _W j `nj x φ? _.W_W 8
3 / * 8 j x nj x j ` ?? φ
( ) _,W_W_W_W /* , j xnj xj xj x j ` cs 8
38 888
(c) Bnjcn, ejm tbus,
3 _W j `n `j x φ?
_.W_W 3 > / *
3 j x n `j x j ` ?? φ ( ) ,_W_W_W * , 6338
5 3 ?+? j xj xj x cs ejm
( ) ._W_W_W_W /* ,
> 3338
8.4 (e) }.86385>={_W ?
j x
5
8
5 ?+?
}5>.8585>.85{ ?
5
8
5 ??
}.3>.8868>.83{ ?
5 ?+?
5 ??
ejm

_.3W_W8 ¾ε? jj x
j
jj km
(c) Bnjcn, ]bnrnakrn,_.W_W3 jjj x j¾ε? _.W_W3 jjj x
j ¾ε?
8
(m) ._W= j
j x ε? Bnjcn, _.W_W == j x j x jj ?ε?ε?
]bnrnakrn,
j nv j x j x j x j x j x j x ε??+?+? _W_)W_W(_)W_W(_W ===
8
5 ==
8
5 ,= ejm
.6_)W_W(_)W_W(_W ==8
5 ==8
5 ,= ??? j x j x j x j x j x km
8.55 ( )._W_W_W 8
5 j x j x j x nv +? ]bus, ( ) _.W_W_W_W
8
5 j x j x j x j x nvnv ?+? Bnjcn, ds
ej nvnj snqunjcn. Ldgnwdsn,
( ) _.W_W_W_W 8

8.58 (e) ]bus,_.W_W_W j x j x ji nvnv? _.W_W_W_W_W_W jij x j x j x j x ji nvnvnvnv ??? Bnjcn,
ds ej nvnj snqunjcn._Wji

(h) ]bus,_.W_W_W j x j x ju km nv? ( ) _.W_W_W_W_W_W juj x j x j x j x ju km nvkm nv ???
Bnjcn, ds ej kmm snqunjcn._Wju

(c) ]bus,_.W_W_W j x j x jv km km ? ( )( )_W_W_W_W_W j x j x j x j x jv km km km km ??

8.53 (e) Sdjcn ds ceusel,_Wj x .6,6_W <? jj x Elsk, .6,6_W :? jj x Jkw,
( )._W_W_W 8
5 j x j x j x nv +? Bnjcn, ( ) _6W_6W_6W_6W
8
.6_,W_W 8

< ? :
5 j x j x j x km ? Bnjcn, ( ) 6_6W_6W_6W
8
.6_,W_W 8
:?

(h) Sdjcn ds ceusel,_Wj y .6,6_W <? jj y Elsk, .6,6_W :? jj y Lnt
wbnrn ejm ern rnel ceusel snqunjcns._,W_W_W j `yj yj y dorn +? _Wj yrn _Wj ydo
Jkw, ( )._W_W_W 8
8
.6_,W_W 8
.
8
.6_,W_W 8
.

8.5= Sdjcn ds ceusel,_Wj x .6,6_W <? jj x Arko tbn sklutdkj ka Vrkhlno 8.53 wn bevn
_.W_W)cks(8 ,6,6 ,65
?

< ? :
?
8.5> (e) wbnrn}{_}W{ j Ej x ε? E ejm ε ern ckoplnx juohnrs wdtb .5<ε Sdjcn akr

jj E ejm ε ern ckoplnx juohnrs wdtb
.5<ε Bnrn, .6,5 ε j j
Bnjcn Ej y _W akr ell veluns ka Bnjcn, ds e
hkujmnm snqunjcn.

(c) wbnrn ejm_W_}W{ jC jb j¾θ? C θ ern ckoplnx juohnrs wdtb .5:θ Sdjcn akr

(m) Sdjcn).cks(=_}W{ jji kτ? =_W ji akr ell veluns ka ds e hkujmnm
snqunjcn.
8

jkt ehsklutnly suooehln.
_}W{ j x
ε

?
√
?
√
ε ? sdjcn

jj
j
j
)(>)(3)( >=3=38=38
GGG +ε+ε+ε++ε+ε+ε++ε+ε+ε+ε?
Jkt akr seln. 4
ε +
ε
ε +
ε
ε +
ε
ε
5
9
5
ε+ε ? Bnjcn, ds ehsklutnly suooehln. _}W{ 3 j x
8.52 (e) _.W 8
5 _W jj x
je ¾? Jkw .8
ds ehsklutnly suooehln._}W{ j x e

++ ? Jkw

8.54 (e) E snqunjcn ds ehsklutnly suooehln da_Wj x ._W √< √
√?j
wn bevn ._W_W_W 8 √<
j x j x j x Bnjcn, ej ehsklutnly suooehln
snqunjcn ds squern suooehln ejm bes tbus adjdtn njnriy.
Jkw ckjsdmnr tbn snqunjcn _.5W_W 5 ¾? jj x
j ]bn ckjvnrinjcn ka ej djadjdtn snrdns cej
hn sbkwj vde tbn djtnirel tnst. Lnt ),( x a ej ? wbnrn e ckjtdjukus, pksdtdvn ejm
mncrnesdji aujctdkj ds akr ell ]bnj tbn snrdns ejm tbn djtnirel
hktb ckjvnrin kr hktb mdvnrin. Akr
.5 x √
?5j je ∧
√√ 6)(lj
5 _W
j jj
j x mkns jkt ckjvnrin. Es e rnsult, _5W_W 5 ¾? jj x
j ds jkt

(h) ]k sbkw tbet ds squern-suooehln, wn khsnrvn tbet bnrn_}W{ j x 8
5
55
55
5
8 ?+
j ds squern-suooehln.
8.85 _.5W cks
_W8 ¾ φ
5cks _W
5 88

^sdji tbn djtnirel tnst (Snn Vrkhlno 8.54, Vert (e) sklutdkj) wn jkw sbkw tbet ds
jkt ehsklutnly suooehln. Jkw,
x c ∧

58
58
58
j x j x j x j x _W_W8_W_Wldo 88
58
5
G
G j km nv x x j x j x
G G x x V V V V +?++?

+√
√
G
+√?+√?+√ ?

?

??
x j x 6
Jkt akr seln. 55

8.83 .56), / 8sdj(_W φ? J j J gjj x Jkw ) / 8(sdj_W 5
6
?
Bnjcn.6?C . 8
_W 8
h N .
,5)58( 58
5 ldo_Wldo
8 5
(h) ]bnj evnrein pkwnr_.W_W8 jj x ¾?
, 8
5
58
.5_W 6
8 88
(c) ]bnj evnrein pkwnr_.W_W3 jjj x ¾?
, 1
(m) ]bnj evnrein pkwnr._W 6 6=
j ` n Ej x
G V
8 = _W
(n) .cks_W 8
φ O
j Ej x Jktn ds e pnrdkmdc snqunjcn. ]bnj evnrein pkwnr_W> j x
.5cks 8
5 cks
5 _W
5 5
5
6
j

?

8.81 Dj necb ka tbn akllkwdji perts, mnjktns tbn aujmeonjtel pnrdkm ejm J r ds e pksdtdvn
djtninr.
>

Eokji ell pksdtdvn sklutdkjs akr J ejm r , tbn soellnst veluns ern ejm>? J .5?r
Bnjcn tbn evnrein pkwnr ds idvnj hy
.2cks= >
J V
>
φ
Eokji ell pksdtdvn sklutdkjs akr J ejm r , tbn soellnst veluns ern ejm56? J .3?r
.>.=cks3 56
J V
(c) Bnrn ejm).9 / 3cks(8_W~ 3 jj x φ? J r oust setdsay tbn rnletdkj .8
9

Eokji ell pksdtdvn sklutdkjs akr ejm J r , tbn soellnst veluns ern ejm5=? J .3?r
.8cks8 5=

(m) Bnrn ejm).3 / >cks(=_W~ = jj x φ? J r oust setdsay tbn rnletdkj .8
3

Eokji ell pksdtdvn sklutdkjs akr ejm J r , tbn soellnst veluns ern ejm1? J .>?r
.2cks= 1
J V
(n) ).> / 3cks(3)> / 8cks(=_W~ > jjj x φ+φ? Yn adrst mntnrodjn tbn aujmeonjtel pnrdkm
ka Bnrn ejm5 J ).> / 8cks( jφ 5 J r oust setdsay tbn rnletdkj .85>
8 r J φ?φ
Eokji ell
pksdtdvn sklutdkjs akr ejm , tbn soellnst veluns ern5 J r >5 ? J ejm Yn jnxt
mntnrodjn tbn aujmeonjtel pnrdkm ka
.5?r
8 J ).> / 3cks( jφ Bnrn ejm r oust setdsay tbn
rnletdkj
3 r J φ?
φ Eokji ell pksdtdvn sklutdkjs akr ejm8 J r , tbn soellnst veluns
ern ejm ]bn aujmeonjtel pnrdkm ka ds tbnj idvnj hy568 ? J .3?r _W~ > j x
.56)56,>(),( 85 ?? LCO J J LCO
Bnjcn tbn evnrein pkwnr ds idvnj hy 8=
6 >
3

(a) ).> / 3cks(3)3 / >cks(=_W~ 1 jjj x φ+φ? Yn adrst mntnrodjn tbn aujmeonjtel pnrdkm
ka Bnrn ejm r oust setdsay tbn rnletdkj5 J ).3 / >cks( jφ 5 J .853
> r J φ?φ
Eokji ell
pksdtdvn sklutdkjs akr ejm , tbn soellnst veluns ern5 J r 15 ? J ejm Yn jnxt
mntnrodjn tbn aujmeonjtel pnrdkm ka
.>?r
8 J ).> / 3cks( jφ Bnrn ejm r oust setdsay tbn
rnletdkj
Eokji ell pksdtdvn sklutdkjs akr ejm r , tbn soellnst veluns
ern ejm ]bn aujmeonjtel pnrdkm ka ds tbnj idvnj hy
8 J
.36)56,1(),( 85 ?? LCO J J LCO
Bnjcn tbn evnrein pkwnr ds idvnj hy 884
6 >
3
3
>5
6
√?g
Jkt akr seln. 5=
._W_W~ ++?+ √
√?g
J gJ j x J j y Suhstdtutdji 5+? g r wn int
? +?+ √
√? Bnjcn ds e pnrdkmdc snqunjcn wdtb e pnrdkm_W~ j y . J
8.82 (e) Jkw ]bn pkrtdkj ka dj tbn rejin.>? J ._>W_W~ +? √
√?j p g j x j x _W~ j x p =6 j ds
idvnj hy }5>=66{_>W_W_>W ?+++ j x j x j x
.=6},59=38{}66666{}68638{ ?++ j
Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ j x p .=6},59=38{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._>W_W~ +? √
√?j p g j yj y _W~ j y p =6 j
}16666{_>W_W_>W ?+++ j yj yj y
.=6},53265>{}66668{}92653{ ?++ j
Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ j y p .=6},53265>{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._>W_W~ +? √
√?j p g jwjw _W~ jw p =6 j
}66666{_>W_W_>W ?+++ jwjwjw
.=6},89565{}6>865{}88366{ ?++ j
Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ jw p .=6},89565{ j
(h) .9? J Jkw ]bn pkrtdkj ka dj tbn rejin._9W_W~ +? √
√?j p g j x j x _W~ j x p 16 j ds
idvnj hy }5>=6666{_9W_W_9W ?+++ j x j x j x
}6666666{}6668638{ ++
.16},5>=8638{ ? j Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ j x p
.16},5>=8638{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._9W_W~ +? √
√?j p g j yj y _W~ j y p 16 j
}1666666{_9W_W_9W ?+++ j x j x j x
}6666666{}6892653{ ++
.16},1892653{ ? j Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ j y p
.16},1892653{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._9W_W~ +? √
√?j p g jwjw _W~ jw p 16 j
}6666666{_9W_W_9W ?+++ jwjwjw
}66666>8{}6588366{ ++

.16},65883>8{ ? j Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ jw p
.16},65883>8{ j
8.84 ).cks(_W~ ρ+τ? j Ej x k
(e) }.55555555{_W~ ?j x Bnjcn .= / ,8 / ,8 φ?ρφ?τ? k E
(h) }.36363636{_W~ ?j x Bnjcn ,3? E ,8 / φ?τk
.8 / φ?ρ
(c) }.311.5311.65311.5311.65{_W~ ?j x Bnjcn ,8? E ,3 / φ?τk
.= / φ?ρ
(m) }.68686868{_W~ ?j x Bnjcn .6,8 / ,8 ?ρφ?τ? k E
8.36 ]bn aujmeonjtel pnrdkm ka e pnrdkmdc snqunjcn wdtb ej ejiuler arnqunjcy J kτ
setdsadns Nq. (8.=9e) wdtb tbn soellnst velun ka ejm . J r
(e) Bnrn Nq. (8.=9e) rnmucns tk.>.6 φ?τk r J φ?φ 8>.6 wbdcb ds setdsadnm wdtb
.5,= ?? r J
(h) Bnrn Nq. (8.=9e) rnmucns tk.2.6 φ?τk r J φ?φ 82.6 wbdcb ds setdsadnm wdtb
.8,> ?? r J
tbds cesn, Nq. (8.=9e) rnmucns tk
5 J ).8.6cks(}Xn{ > / jn
j ` φ?φ
55 88.6 r J φ?φ wbdcb ds setdsadnm wdtb .5,56 55 ?? r J
Yn jnxt mntnrodjn tbn aujmeonjtel pnrdkm ka Dj tbds
cesn, Nq. (8.=9e) rnmucns tk
8 J ).5.6sdj(Do{ 56 / j `n
j ` φ?φ
88 85.6 r J φ?φ wbdcb ds setdsadnm wdtb .5,86 88 ?? r J
Bnjcn tbn aujmeonjtel pnrdkm ka ds idvnj hy J _W~ j x c
.86)86,56(),( 85 ?? LCO J J LCO
(m) Yn adrst mntnrodjn tbn aujmeonjtel pnrdkm ka Dj tbds cesn, Nq.
(8.=9e) rnmucns tk
5 J ).3.5cks(3 jφ
55 83.5 r J φ?φ wbdcb ds setdsadnm wdtb .53,86 55 ?? r J Yn jnxt
mntnrodjn tbn aujmeonjtel pnrdkm ka8 J ).>.6>.6sdj(= φ+φj Dj tbds cesn, Nq. (8.=9e)
rnmucns tk wbdcb ds setdsadnm wdtb88 8>.6 r J φ?φ .5,= 88 ?? r J Bnjcn tbn
aujmeonjtel pnrdkm ka ds idvnj hy J _W~ = j x .86)=,86(),( 85 ?? LCO J J LCO
(n) Yn adrst mntnrodjn tbn aujmeonjtel pnrdkm ka Dj tbds
cesn, Nq. (8.=9e) rnmucns tk
5 J ).9>.6>.5cks(> φ+φj
55 8>.5 r J φ?φ wbdcb ds setdsadnm wdtb .3,= 55 ?? r J Yn
jnxt mntnrodjn tbn aujmeonjtel pnrdkm ka8 J ).1.6cks(= jφ Dj tbds cesn, Nq. (8.=9e)
rnmucns tk wbdcb ds setdsadnm wdtb88 81.6 r J φ?φ .3,56 88 ?? r J Yn adjelly
mntnrodjn tbn aujmeonjtel pnrdkm ka3 J ).>.6sdj( jφ Dj tbds cesn, Nq. (8.=9e) rnmucns
tk wbdcb ds setdsadnm wdtb33 8>.6 r J φ?φ .5,= 33 ?? r J Bnjcn tbn aujmeonjtel pnrdkm
J ka ds idvnj hy_W~ > j x .86)=,56,=(),,( 385 ?? LCO J J J LCO
8.35 ]bn aujmeonjtel pnrdkm J ka e pnrdkmdc snqunjcn wdtb ej ejiuler arnqunjcy kτ
setdsadns Nq. (8.=9e) wdtb tbn soellnst velun ka J ejm .r
Jkt akr seln. 51

(e) Bnrn Nq. (8.=9e) rnmucns tk.1.6 φ?τk r J φ?φ 81.6 wbdcb ds setdsadnm wdtb
.3,56 ?? r J

(h) Bnrn Nq. (8.=9e) rnmucns tk.82.6 φ?τk r J φ?φ 882.6 wbdcb ds setdsadnm wdtb
.9,>6 ?? r J

(c) Bnrn Nq. (8.=9e) rnmucns tk.=>.6 φ?τk r J φ?φ 8=>.6 wbdcb ds setdsadnm wdtb
.4,=6 ?? r J

(m) Bnrn Nq. (8.=9e) rnmucns tk.>>.6 φ?τk r J φ?φ 8>>.6 wbdcb ds setdsadnm wdtb
.55,=6 ?? r J

(n) Bnrn Nq. (8.=9e) rnmucns tk.1>.6 φ?τk r J φ?φ 81>.6 wbdcb ds setdsadnm wdtb
.53,=6 ?? r J

8.38 Bnrn Nq. (8.=9e) rnmucns tk.62.6 φ?τk r J φ?φ 862.6 wbdcb ds setdsadnm wdtb
Akr e snqunjcn.5,8> ?? r J )sdj(_W~ 88 jj x τ? wdtb e aujmeonjtel pnrdkm ka 8>? J ,
Nq. (8.=9e) rnmucns tk .88> 8 r φ?τ Akr nxeopln, akr 8?r wn bevn
Ejktbnr snqunjcn wdtb tbn seon aujmeonjtel pnrdkm ds khtedjnm
hy snttdji wbdcb lnems tk
.51.68> / =8 φ?φ?τ 3?r .8=.68> / 13 φ?φ?τ ]bn ckrrnspkjmdji pnrdkmdc
snqunjcns ern tbnrnakrn )51.6sdj(_W~ 8 jj x φ? ejm ).8=.6sdj(_W~
3 jj x φ?
8.33 ]bn tbrnn pereontnrs ejm,, k E ρ ka tbn ckjtdjukus-tdon sdijel cej hn
mntnrodjnm arko
)(t x e
)cks()(_W ρ+?? j] Ej] x j x ke hy snttdji 3 mdstdjct veluns ka
Akr nxeopln
,sdj)sdj(cks)cks()cks(_5W θ?ρ+ρ?ρ+? ] E] E] E x kkk ,
.sdj)sdj(cks)cks()cks(_5W γ?ρρ?ρ+? ] E] E] E x kkk
Suhstdtutdji tbn adrst nquetdkj djtk tbn lest twk nquetdkjs ejm tbnj emmdji tbno wn int
ε γ+θ
? 8
)cks( ] k wbdcb cej hn sklvnm tk mntnrodjn k . Jnxt, arko tbn snckjm
]
)cks( tej
wbdcb cej hn sklvnm tk mntnrodjn .ρ Adjelly, tbn pereontnr ds mntnrodjnm arko tbn adrst
nquetdkj ka tbn lest pein.
Jkt akr seln. 59
k] ]
? φ
? Dj tbds cesn θ?ρ+φ? )cks(_W j Ej x ejm
( ) θ?ρ+φ?ρ+φ+?+ )cks()5(cks_5W j Ej Ej x . Sdjcn ell seopln veluns ern nquel, tbn
tbrnn pereontnrs cejjkt hn mntnrodjnm ujdqunly.
Adjelly ckjsdmnr tbn cesn .8 8
k] ]
< φ
? Dj tbds cesn )cks(_W ρ+? j] Ej x k
doplydji)cks( ρ+τ? j E k .φ:?τ ] kk Es nxpledjnm dj Snctdkj 8.8.5, e mdidtel
sdjuskdmel snqunjcn wdtb ej ejiuler arnqunjcy kτ irnetnr tbej essuons tbn dmnjtdty
ka e sdjuskdmel snqunjcn wdtb ej ejiuler arnqunjcy dj tbn rejin . Bnjcn,
cejjkt hn ujdqunly mntnrodjnm arko
φ .6 φ<τ
k )cks(_W ρ+? j] Ej x k .
8.3= Da ds pnrdkmdc wdtb e pnrdkm , tbnj).cks(_W j] j x k? _Wj x J
( ) ).cks(_Wcks_W 666 j] j x J] j] J j x ??+?+ ]bds dopldns r J] k φ? 8 wdtb
r ejy jkjznrk pksdtdvn djtninr. Bnjcn tbn seopldji retn oust setdsay tbn rnletdkj
Da d.n.,. / 8 J r ] kφ? ,86?k ,2 / φ?] tbnj wn oust bevn r J φ? φ
8 2
86 . ]bn
soellnst velun ka ejm r setdsaydji tbds rnletdkj ern J =? J ejm ]bn
aujmeonjtel pnrdkm ds tbus
.>?r
=? J .
8.3> (e) Akr ej djput tbn kutput ds,8,5_,W ?dj x d
.8,5_,8W_5W_8W_5W_W_W 85856 ?++++? dj yej yej x hj x hj x hj y dddddd ]bnj, akr
ej djput tbn kutput ds_,W_W_W 85 j Hx j Ex j x +? _)W_W(_W 856 j Hx j Ex hj y +?
_)5W_5W(_)8W_8W(_)5W_5W( 855858855 ++++++ j Hyj Eyej Hx j Ex hj Hx j Ex h
_)8W_8W( 855 ++ j Hyj Eye _5W_8W_5W_W( 55585556 +++? j yej x hj x hj x h E
_)8W_5W_8W_5W_W(_)5W 888583888588 ++++++ j x ej x ej x hj x hj x h Hj ye
_.W_W 85 j Hyj Ey +? Bnjcn, tbn systno ka Nq. (8.52) ds ldjner.
(h) Akr ej djput tbn kutput ds,8,5_,W ?dj x d µµ??
ktbnrwdsn.,6
d
Akr ej djput _,W_W_W 85 j Hx j Ex j x +? tbn kutput akr G,8,,6 L Lj µµ? ds
_W_W_ / W_ / W_ / W_W 8585 j Hyj Ey Lj Hx Lj Ex Lj x j y +?+?? . Akr ell ktbnr veluns ka

(c) Akr ej djput tbn kutput ds,8,5_,W ?dj x d .8,5_, / W_W ?? d O j x j y dd ]bnj, akr ej djput
tbn kutput ds_,W_W_W 85 j Hx j Ex j x +? _.W_W_ / W_ / W_W 8585 j Hyj Ey O j Hx O j Ex j y +?+?
Bnjcn tbn systno ka Nq. (8.85) ds ldjner.
Jkt akr seln. 52
?
g dd dg j x
O j y ]bnj, akr
?
j y
_.W_W_W 5
_W 5
O Hg j x
? Bnjcn tbn systno ka
Nq. (8.15) ds ldjner.
(n) ]bn adrst tnro kj tbn XBS ka Nq. (8.1>) ds tbn kutput ka e aectkr-ka-8 up-seoplnr.
]bn snckjm tnro kj tbn XBS ka Nq. (8.1>) ds sdoply tbn kutput ka ej ujdt mnley akllkwnm hy e aectkr-ka-8 up-seoplnr, wbnrnes, tbn tbdrm tnro ds tbn kutput ka ej ujdt
emvejcn kpnretkr akllkwnm hy e aectkr-ka-8 up-seoplnr. Yn bevn sbkwj dj Vert (h) tbet
tbn up-seoplnr ds e ldjner systno. Okrnkvnr, tbn ujdt mnley ejm tbn ujdt emvejcn kpnretkr ern ldjner systnos. E cescemn ka twk ldjner systnos ds ldjner ejm tbn ldjner
ckohdjetdkj ka ldjner systnos ds elsk ldjner. Bnjcn, tbn aectkr-ka-8 djtnrpkletkr ka Nq.
(8.1>) ds e ldjner systno.
(a) Akllkwdji tbn eriuonjts idvnj dj Vert (n), wn cej sdodlerly sbkw tbet tbn aectkr-ka-
3 djtnrpkletkr ka Nq. (8.11) ds e ldjner systno.
8.31 (e) Akr ej djput_.W_W 3 j x jj y ? ,8,5_,W ?dj x d tbn kutput ds
]bnj, akr ej djput
.8,5_,W_W 3 ?? dj x jj y dd
_,W_W_W 85 j Hx j Ex j x +? tbn kutput ds
Bnjcn tbn systno ds ldjner.
( )_W_W_W 85 3 j Hx j Ex jj y +?
_.W_W 85 j Hyj Ey +?
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Es
akr ejm tbn systno ds ceusel.
_,W_W jj x ν? _.W_W 3 jjjb ν?
6_W ?jb ,6<j
Lnt 5 akr ell veluns ka ]bnj_W ?j x .j 3 _W jj y ? ejm es Sdjcn e
hkujmnm djput rnsults dj ej ujhkujmnm kutput, tbn systno ds jkt HDHK stehln.
√_Wj y .√j
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly. Da_Wj y _W5 j y _Wj x _,W5 j x
tbnj Bkwnvnr,_W_W5 kjj x j x ? _.W_W_W 3 5
3 5 kjj x jj x jj y ?? ? _W kjj y
Sdjcn_.W)( 3
kk jj x jj _,W_W5 kjj yj y tbn systno ds jkt tdon-djverdejt.
(h) Akr ej djput._)W(_W >j x j y ? ,8,5_,W ?dj x d tbn kutput ds .8,5,_)W(_W
> ?? dj x j y dd
]bnj, akr ej djput _,W_W_W 85 j Hx j Ex j x +? tbn kutput ds
Bnjcn tbn systno ds jkjldjner.
( )> 85 _W_W_W j Hx j Ex j y +?
._)W(_)W( >
8 >
akr ejm tbn systno ds ceusel.
_,W_W jj x ν? ._)W(_W >jjb ν?
6_W ?jb ,6<j
Jkt akr seln. 54

Akr e hkujmnm djput ,_W √< Hj x tbn oeijdtumn ka tbn kutput seoplns ern
._W_)W(_W >>> √<?? Hj x j x j y Es tbn kutput ds elsk e hkujmnm snqunjcn, tbn
systno ds HDHK stehln.
_W_W5 kjj x j x ? tbnj ( ) _.W_W_)W(_W >>
55 kk jj yjj x j x j y ??? Bnjcn, tbn systno
ds tdon-djverdejt.
_W_W l
lj x j y θ e jkjznrk ckjstejt. Akr ej djput tbn
kutput ds ]bnj, akr ej djput
tbn kutput ds
l _,W_W_W 85 j Hx j Ex j x +?
( ) + +θ? ++θ? ???
lll
llll j Hx j Ex j Hx j Ex j y
_.W_W 85 j Hyj Ey + Bnjcn tbn systno ds jkjldjner.
akr tbn systno ds jkjceusel.
_,W_W jj x ν? ._W_W 6
ν+θ? √
?l ljjb
6_W jb ,6<j
.=_W √<+θ Hj y Es tbn kutput ds elsk e hkujmnm snqunjcn, tbn systno ds HDHK
stehln.
_W_W5 kjj x j x ? tbnj Bnjcn, tbn systno ds
tdon-djverdejt.
?l l

(m) ( ._W8lj_W j x j y +? ) Akr ej djput ,8,5_,W ?dj x d tbn kutput ds ( ),_W8lj_W j x j y dd +?
.8,5?d ]bnj, akr ej djput _,W_W_W 85 j Hx j Ex j x +? tbn kutput ds
( ) _.W_W_W_W8lj_W 8585 j Hyj Eyj Hx j Ex j y +++? Bnjcn tbn systno ds jkjldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn_,W_W jj x ν? ( )_W8lj_W jjb ν++ .
Akr Bnjcn, tbn systno ds jkjceusel..6)8lj(_W,6 ?< jbj
( ) .8lj_W √<+ Hj y Es tbn kutput ds elsk e hkujmnm snqunjcn, tbn systno ds HDHK
stehln.
Jkt akr seln. 86

_W_W5 kjj x j x ? tbnj ( ) _.W_W8lj_W5 kk jj yjj x j y ?+? Bnjcn, tbn systno ds
tdon-djverdejt.
(n) wdtb e jkjznrk ckjstejt. Akr ej djput tbn kutput
ds ]bnj, akr ej djput
_,8W_W +ε? j x j y ,8,5_,W ?dj x d
_,8W_W +ε? j x j y dd .8,5?d _,W_W_W 85 j Hx j Ex j x +? tbn kutput ds
_.W_W_8W_8W_W 8585 j Hyj Eyj x Hj x Ej y +?+ε++ε? Bnjcn tbn systno ds ldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Akr_,W_W jj x ν? _.8W_W +εν? jjb
,6<j .6_W ?jb Bnjcn, tbn systno ds ceusel.
._W √<ε? Hj y Es tbn kutput ds elsk e hkujmnm snqunjcn, tbn systno ds HDHK stehln.
_W_W5 kjj x j x ? tbnj _.W_8)(W_8W_W 55 kk jj yjj x j x j y ?+ε?+ε? Bnjcn, tbn
systno ds tdon-djverdejt.
(a) Akr ej djput_.=W_W ? j x j y ,8,5_,W ?dj x d tbn kutput ds .8,5_,=W_W ?? dj x j y dd
]bnj, akr ej djput _,W_W_W 85 j Hx j Ex j x +? tbn kutput ds _=W_=W_W 85 +? j Hx j Ex j y
_.W_W 85 j Hyj Ey +? Bnjcn tbn systno ds ldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Akr_,W_W jj x ν? _.=W_W ν? jjb ,6<j
.6_W ?jb Bnjcn, tbn systno ds ceusel.
._W √<? Hj y Es tbn kutput ds elsk e hkujmnm snqunjcn, tbn systno ds HDHK stehln.
_W_W5 kjj x j x ? tbnj _.W_=W_W5 kk jj yjj x j y ?? Bnjcn, tbn systno ds tdon-
djverdejt.
8.39 Lnt ejm hn tbn kutputs ka e onmdej adltnr ka lnjitb akr djputs
ejm , rnspnctdvnly. Da
_Wj y _W5 j y 58 +G _Wj x
_W5 j x _W_W5 kjj x j x ? , tbnj
}_W,_,5W_,W_,5W,_,W{onm_W 555555 G j x j x j x j x G j x j y ++? GG
}_W,_,5W_,W_,5W,_,W{onm G jj x jj x jj x jj x G jj x kkkkk ++? GG
_.W kjj y ? Bnjcn, tbn systno ds tdon-djverdejt.
8.32 _.5W_W8_5W_W ++? j x j x j x j y Akr ej djput ,8,5_,W ?dj x d tbn kutput ds
.8,5_,5W_W8_5W_W ?++? dj x j x j x j y dddd ]bnj, akr ej djput _,W_W_W 85 j Hx j Ex j x +?
tbn kutput ds
_5W_5W_W8_W8_5W_5W_W 858585 +++++? j Hx j Ex j Hx j Ex j Hx j Ex j y
_.W_W 85 j Hyj Ey +? Bnjcn tbn systno ds ldjner.
Da tbnj_,W_W5 kjj x j x ? _.W_5W_W8_5W_W5 kkkk jj yjj x jj x jj x j y ?++?
Bnjcn, tbn systno ds tdon-djverdejt.
Jkt akr seln. 85

]bn dopulsn rnspkjsn ka tbn systno ds _5W_W8_5W_W ν+ν+ν? jjjjb . Jkw
.5_6W_5W ?ν?b Sdjcn 6_W jb akr ell veluns ka ,6<j tbn systno ds jkjceusel.
8.34 Akr ej djput_.5W_5W_W_W 8 +? j x j x j x j y ,8,5_,W ?dj x d tbn kutput ds
.8,5_,5W_5W_W_W 8 ?+? dj x j x j x j y dddd ]bnj, akr ej djput tbn
kutput ds
( ) ( )( )_5W_5W_5W_5W_W_W_W 8585 8
85 +++++? j Hx j Ex j Hx j Ex j Hx j Ex j y
Bnjcn tbn systno ds jkjldjner._.W_W 85 j Hyj Ey +
Da tbnj_,W_W5 kjj x j x ? _5W_5W_W_W 55 8 55 +? j x j x j x j y
_5W_5W_W 8 +? kkk jj x jj x jj x _.W kjj y ? Bnjcn, tbn systno ds tdon-
djverdejt.
akr ell veluns ka
6_W ?jb ,6<j tbn systno ds ceusel.
8.=6 . _5W
+? j y
j x j yj y Jkw akr ej djput _,W_W jj x ε¾? tbn kutput
ckjvnrins tk skon ckjstejt
G es .√j ]bn djput-kutput rnletdkj ka tbn systno es
rnmucns tk√j
G
.ε?G
Akr ej djput tbn kutput ds,8,5_,W ?dj x d .8,5, _5W
_W _5W
d
d dd ]bnj,
akr ej djput _,W_W_W 85 j Hx j Ex j x +? tbn kutput ds . _5W
_W_W _5W
Kj tbn ktbnr bejm,
_5W
Da tbnj_,W_W5 kjj x j x ? _.W _5W
_W _5W

8.=5 _.5W_5W_W_W 8 +? j yj yj x j y
Akr ej djput tbn kutput ds
]bnj, akr ej djput
,8,5_,W ?dj x d .8,5_,5W_5W_W_W 8 ?+? dj yj yj x j y dddd
_,W_W_W 85 j Hx j Ex j x +? tbn kutput ds
_.5W_5W_W_W_W 8
85 ++? j yj yj Hx j Ex j y Kj tbn ktbnr bejm, _W_W 85 j Hyj Ey +
Jkt akr seln. 88
_.W_5W_5W_W_5W_5W_W 8 8 885
8 55 j yj Hyj Hyj Hx j Eyj Eyj Ex +++? Bnjcn tbn
systno ds jkjldjner.
Lnt hn tbn kutput akr ej djput , d.n., ]bnj, akr ej
djput tbn kutput ds idvnj hy , kr
dj ktbnr wkrms, tbn systno ds tdon-djverdejt.
_Wj y _Wj x _.W_W_W_W 5 8 +? j yj yj xj y

Jkw, akr ej djput _W_W jj x ε¾? , tbn kutput ckjvnrins tk skon ckjstejt G es_Wj y √j .
]bn mdaanrnjcn nquetdkj mnscrdhdji tbn systno es √j rnmucns tk kr
d.n.,
,ε?8 G .ε? G
8.=8 ]bn dopulsn rnspkjsn ka tbn aectkr-ka-3 djtnrpkletkr ka Nq. (8.11) ds tbn kutput akr ej
djput ejm ds idvnj hy_W_W jj x u ν?
_)8W_8W(_)5W_5W(_W_W 3
8 +ν+ν++ν+ν+ν? jjjjjjb kr nqudvelnjtly hy
.88,,,5,_}W{ 3
5 ? jjb
8.=3 ]bn djput-kutput rnletdkj ka e aectkr-ka- L djtnrpkletkr ds idvnj hy
( ._W_W_W_W 5
L
? ) Dts dopulsn rnspkjsn ds tbn kutput akr
ej djput ejm ds tbus idvnj hy_W_W jj x u ν? ( )_W_W_W_W 5
5

8.== ]bn dopulsn rnspkjsn ka e ceusel mdscrntn-tdon systno setdsadns tbn mdaanrnjcn
nquetdkj
_Wjb
_.W_5W_W jjbejb ν? Sdjcn tbn systno ds ceusel, wn bevn akr
Nveluetdji tbn ehkvn mdaanrnjcn nquetdkj akr
6_W ?jb
.6<j ,6?j wn errdvn et
ejm tbus5_5W_6W ? ebb .5_6W ?b Jnxt, akr 5?j , wn bevn ejm tbus
Ckjtdjudji wn int akr
wdtb Arko tbn mdaanrnjcn nquetdkj wn tbnj bevn
, d.n., Sdjcn tbn lest nquetdkj bklms akr
hy djmuctdkj, dt bklms akr
5 _5W ? jejb .6:j
6_5W_W ? jbejb ._5W_W jejebjb ??

8.=> Es ejm ern rdibt-sdmnm snqunjcns, essuon_Wj x _Wjb 6_W ?j x akr ell ejm
ejm Bnjcn,
5 J j < 6_W ?jb .8 J j < 6_WK_W_W * ?? j x jbj y akr ell ejm tbus85 J J j +<
Jkt akr seln. 83
8585
? √
+?
√
+?
? ? ? √
+?
√
?
√
?
√
+?
√
+?
√
? 8588 8585 8
_W_W_W_W_W_W J J j J g J g J J j J J j J g
g j x g bg j x g bg j x g b
es?? √
? 58858
_W_W_W_W J o J g g J J o J g
o x g bo x g b 6_W ?o x akr ell . Bnjcn,5 J o <
._W_W_W
√
?
√
√? ,,
j g
√
?
√
j g

8.=9 Jkw arko Nq. (8.98) ej erhdtrery djput cej hn nxprnssnm es
wbdcb cej hn rnwrdttnj usdji Nq. (8.=5h) es
_Wj x
( ) ¾¾? √
√?g
√?g
√?g
g jg x
Sdjcn ds tbn rnspkjsn ka ej L]D systno akr ej djput_Wjs _,Wj¾ ds tbn rnspkjsn
akr ej djput ejm
_W g js _W g j ¾ _5W g js ds tbn rnspkjsn akr ej djput Bnjcn,
tbn kutput akr ej djput ds idvnj hy
_.5W ¾ g j
√?g
√?
√
√? jsj x jsj x g jsg x g jsg x j y
g g
._W~_W_W ? √
oj x obj y
? ?+?+ √
√?
√
. J
8.=4 Dj tbds prkhlno wn oegn usn ka tbn dmnjtdty _.W_WK_W * r ojr joj ν?νν
Jkt akr seln. 8=

(e) ( ) ( )_5W8_W=_8WK_5W8_8W3_WK_W_W *5*55 ν+ν++ν+νν?? jjjjjjbj x j y
_5WK_8W1_WK_8W58_8WK_8W3 *** νννν++νν? jjjjjj _8WK_5W8 * +ν+ν+ jj
_5WK_5W=_WK_5W2 ** ν+ν+ν+ν jjjj . Bnjcn
_W=_5W2_3W8_3W1_8W58_W3_W5 jjjjjjj y ν++ν+ν+νν+ν?
_.3W1_8W58_W_5W2_3W8 νν+ν++ν+ν? jjjjj
(h) ( ) ( )_5W_8W>.5_=W3K_5W8_3W>_WK_W_W *8*88 +νν+ν+ν+ν?? jjjjjjbj x j y
_=WK_5W1_5WK_3W>_8WK_3W>.9_=WK_3W5> **** ν+ν++νννν+νν? jjjjjjjj
_5WK_5W8_8WK_5W3 ** +ν+νν+ν+ jjjj _8W>_>W>.9_9W5> νν+ν? jjj
_.8W8_5W3_3W1 +νν+ν+ jjj
(c) ( ) ( )_5W_8W>.5_=W3K_5W8_8W3_WK_W_W *8*53 +νν+ν+νν?? jjjjjjbj x j y
_=WK_5W1_5WK_8W3_8WK_8W>.=_=WK_8W4 **** ν+ν+νννν+νν? jjjjjjjj
_5WK_5W8_8WK_5W3 ** +ν+ν+ν+ν jjjj _5W3_=W>.=_1W4 νν+ν? jjj
_8W8_5W3_5W3_3W1 +ν+ννν jjjj _3W1_5W1_8W8 νν+ν? jjj
_.1W4_=W>.= ν+ν+ jj

(m) ( ) ( )_5W8_W=_8WK_5W8_3W>_WK_W_W *5*8= νν++ν+ν+ν?? jjjjjjbj x j y
_8WK_5W8_5WK_3W56_WK_3W86_8WK_3W> **** +ν+ννννν++νν? jjjjjjjj
_5WK_5W=_WK_5W2 ** ν+νν+ν+ jjjj _3W8_=W56_3W86_5W> +ννν+ν? jjjj
_W=_5W2 jj ν+ν+ _.=W56_3W86_5W>_W=_5W2_3W8 νν+νν+ν++ν? jjjjjj
8.>6 (e) _WK_W_W * j yj x ju ?
.2=},=,5=,88,59,=8,8>,11,83,=>,86,>,=8,8={ ? j
(h) _WK_W_W * jwj x jv ?
.555},56,=,5>,1,53,36,82,3,51,56,>,9,58{ ? j
(c) _WK_W_W * j yjwji ?
Jkt akr seln. 8>
akr ds mnadjnm akr
85 O oj O
_W,5 ojb J o ? 855 O J j O , kr nqudvelnjtly, akr
. Ldgnwdsn, akr5855 J O j J O ++ _W,8 ojb J o ? ds mnadjnm akr
, kr nqudvelnjtly, akr885 O J j O .8885 J O j J O ++ Akr tbn spncdadnm
snqunjcns .1,8,=,3 8585 ???? O O J J (e) ]bn lnjitb ka ds_Wj y
585)3(8=155588 ?++?++ J O J O . (h) ]bn rejin ka akrj 6_W j y ds
),oex(),odj( 88558855 J O J O j J O J O ++++ , d.n.,
Akr tbn spncdadnm snqunjcns tbn rejin ka ds.8855 J O j J O ++ j .565 j
8.>8 Jkw,
√?g
?? √
√?g
J g x g J j x J j x J j x j _.W_W_WK_W_W 885588*55v .8 o J g ?
]bnj ?? √
√?o

8.>3 _WK_W_WK_WK_W_W 3*3*8*5 j x j yj x j x j x ji ?? wbnrn _.WK_W_W 8*5 j x j x j y ? Jkw
_.WK_W_W 88*55 J j x J j x jv ? Mnadjn _.WK_W_W 33* J j x jvjb ? ]bnj arko tbn
rnsults ka Vrkhlno 8.>8, _.W_W 85 J J j yjv ? Bnjcn,
_.WK_W_W 33*85 J j x J J j yjb ? ]bnrnakrn, oegdji usn ka tbn rnsults ka Vrkhlno
8.>8 eiedj wn int _.W_W 385 J J J j yjb ?
8.>= Suhstdtutdji hy ?? √
√?g
g bg j x jbj x j y _.W_W_WK_W_W * g oj dj tbds nxprnssdkj, wn
int Bnjcn tbn ckjvklutdkj kpnretdkj ds
ckooutetdvn.
Lnt ( ) ( +?+? √
√?g
g bg bg j x jbjbj x j y _W_W_W_W_WK_W_W 8585* )
Bnjcn tbn
+ ?+? √
√?
√
√?g g

snqunjcn, tbn rnsult ka tbds ckjvklutdkj cejjkt hn mntnrodjnm. Hut
_).WK_W(K_W_WK_WK_W 5*8*35*8*3 j x j x j x j x j x j x ? _WK_W 5*8 j x j x
Jkt akr seln. 81

_).WK_W(K_W_WK_WK_W 5*3*85*3*8 j x j x j x j x j x j x ? Jkw akr ell veluns
ka , ejm bnjcn tbn kvnrell rnsult ds znrk. Es e rnsult, akr tbn idvnj snqunjcns
6_WK_W 5*3 ?j x j x
j

g jbg x jbj x j y _W_W_WK_W_W * ejm
._W_W_WK_W_W * ?? g
g jbg ijijbj a Ckjsdmnr _WK_)WK_W(_W **5 jijbj x jw ?
_.W_W_W_WK_W * g ojbg x oijij y g o
?? Jkw ckjsdmnr _)WK_W(K_W_W **8 jijbj x jw ?
_.W_W_W_WK_W * og jboig x j a j x og
?? ]bn mdaanrnjcn hntwnnj tbn

dopldns ]bus,

< ?
g ojbg x oi 66
_.W_W_W Ell suos bevn kjly e adjdtn juohnr ka tnros. Bnjcn,
tbn djtnrcbejin ka tbn krmnr ka tbn suooetdkjs ds ùstdadnm ejm wdll idvn ckrrnct rnsults.
H) Essuoptdkjs0 ejm ern stehln snqunjcns, ejm ds e hkujmnm snqunjcn
wdtb
5 _W_W_W_W_W g
g g g g o x g bg o x g bo y√< Hj x _W
_W 85 , og g ξ+ wdtb ._W
85 , Ho jg g ξξ Dj tbds cesn, ell suos bevn naanctdvnly kjly e adjdtn
juohnr ka tnros ejm tbn nrrkr _W 85 , og g ξ cej hn rnmucnm hy cbkksdji ejm
suaadcdnjtly lerin. Es e rnsult, dj tbds cesn tbn prkhlno ds eiedj naanctdvnly rnmucnm
tk tbet ka tbn kjn-sdmnm snqunjcns. ]bus, tbn djtnrcbejin ka tbn krmnr ka tbn
suooetdkjs ds eiedj ùstdadnm ejm wdll idvn ckrrnct rnsults.
5g
8g
Bnjcn, akr tbn ckjvklutdkj tk hn esskcdetdvn, dt ds suaadcdnjt tbet tbn snqunjcns hn stehln
ejm sdjiln-sdmnm.
8.>9 Sdjcn ds ka lnjitb._W_W_W ? √ √?g g bg j x j y _Wg b O ejm mnadjnm akr ,56 O g
tbn ckjvklutdkj suo rnmucns tk wdll hn jkjznrk akr
ell tbksn veluns ka ejm akr g wbdcb
._W_W_W )5( 6 ?
g g bg j x j y _Wj y
j g j setdsadns .56 J g j Odjdouo
velun ka ejm kccurs akr lkwnst et6? g j j 6?j ejm .6?g Oexdouo velun ka
ejm kccurs akr oexdouo velun ka et5? J g j g .5 O ]bus 5? O g j

8.>2 ]bn oexdouo velun ka kccurs et wbnj ell
prkmuct tnros ern prnsnjt. ]bn oexdouo velun ds idvnj hy
._W_W_W 5 6 ?
? J g
g x g j x j y _Wj y 5? J j
._5W 5 6 5?

8.>4 ]bn oexdouo velun ka kccurs et wbnj ell
prkmuct tnros ern prnsnjt. ]bn oexdouo velun ds idvnj hy
._W_W_W 5 6 ?
? J g
g bg j x j y _Wj y 5? J j
._5W 5 6 5?

?? √ √?g nvnvnvnv g ig jbjbjij y _.W_W_WK_W_W *
? √ √?g nvnv g ig jbj y _.W_W_W g .g ]bnj tbn suooetdkj kj tbn lnat
hnckons ?+? √ √?
√ √?g g nvnvnvnv g ig jbg ig jbj y _W)_(W_W_W_W
_.Wj y? Bnjcn ds ej nvnj snqunjcn._WK_W * jbji nvnv
+? ? √ √?
√ √?
Bnjcn ds ej kmm snqunjcn. _WK_W * jbji km nv
+? ? √ √?
√ √?
Bnjcn ds ej nvnj snqunjcn. _WK_W * jbji km km
8.15 ]bn dopulsn rnspkjsn ka tbn cescemn ds idvnj hy _WK_W_W 8*5 jbjbjb ? wbnrn
ejm Bnjcn,_W_W5 jjb j¾ε? _.W_W8 jjb
j¾θ? ( _.W_W 6
jjb j g
g jg ¾ θε? ?
8.18 Jkw ]bnrnakrn_.W_W jjb j¾ε? _W_W_W_W 6 g j x g j x g bj y g
g g ε? ? √
_.5W_W_5W_W_W_W 65 ε+? εε+? ε+? √ ?
√ ? j yj x g j x j x g j x j x g
g g
g
Bnjcn, _.5W_W_W ε? j yj yj x ]bus tbn djvnrsn systno ds idvnj hy
]bn dopulsn rnspkjsn ka tbn djvnrsn systno ds idvnj hy_.5W_W_W ε? j x j x j y
.56},,5{_W ε? jjb
8.13 Arko tbn rnsults ka Vrkhlno 8.18 wn bevn ( _.W_W 6
jjb j g
Jkt akr seln. 82
_W_W_W_W_W_W 6 66
g j x g j x og bg j x j y g
g
o
g
o
_5W_W_5W_W_W 6
√
?
+
?
+√
?
+
?
+ r j x j x r j x j x j y r
r r
r
r
r
o
or o
._3W_8W_5W_5W_W 38
G+ε+ε+ε+θ+? j x j x j x j yj x ]bn djvnrsn systno ds
tbnrnakrn idvnj hy _.8W_5W)(_W_W εθ+θ+ε? j yj yj yj x
8.1= (e) _.WK_W_WK_W_WK_WK_WK_W_W =*38*53*3*8*5 jbjbjbjbjbjbjbjbjb ++?
(h) . _WK_WK_W5
_)8W8_5W(K_)5W3_8W8(_WK_W *8*5 +ν+ν+νν? jjjjjbjb
_WK_W_WK_W_WK_W_WK_W **** 88538888553588 +ν+ν+νν+ν+ννν? jjjjjjjj
? _W_W_W_W 31=338 +νν+νν jjjj ejm
_.W_W_W_W_W_W 535839>>3 +ν+νν+ν+ν? jjjjjjb ]bnrnakrn,
_W_W_W_W_W_W_W_W_W_W 535839>>31=338 +ν+νν+ν+ν++νν+νν? jjjjjjjjjjb
_.W_W_W_W_W 31535834>> +ν+ν+ν+ν+ν? jjjjj

? ?
j x wn
(h) ]bn lnjitb ka ds_Wj x .=5=9 ?+ ^sdji

? ?
j x wn
(c) ]bn lnjitb ka ds_Wj x .=5>2 ?+ ^sdji

? ?
=
j x wn
Jkt akr seln. 84

8.19 _.W_5W_W jhx jeyj y +? Bnjcn, _.6W_5W_6W hx ey y +? Jnxt,
Ckjtdjudji aurtbnr dj e sdodler
wey wn khtedjnm
._W_5W_W 6
5 +? ? + j
(e) Lnt hn tbn kutput mun tk ej djput ]bnj
Da
._W_5W_W 6 5
5 5 +? ?
+ j g
g jj g hx e yej y _,W_W5 kjj x j x ? tbnj
Bkwnvnr,
g jj r hx e yejg hx e yej y
._W_5W_W_5W_W 6 6
k kk r hx e yejg hx e yejj y
Bnjcn da_W_W5 kjj yj y ,6_5W y d.n., tbn systno ds tdon-verdejt. ]bn systno ds
tdon-djverdejt da ejm kjly da ,6_5W ? y es tbnj _W_W5 kjj yj y ? .
(h) Lnt ejm hn tbn kutputs mun tk djputs ejm , rnspnctdvnly.
Lnt hn tbn kutput mun tk ej djput
_W5 j y _W5 j y _W5 j x _W5 j x
_Wj y _.W_W 85 j x j x θ+ε Bkwnvnr, ?θ+ε _W_W 85 j yj y
wbnrnes,,_W_W_5W_5W 6 86 5
? ++ j
g g jjj g hx eg hx e ye ye
Bnjcn, tbn systno ds
._W_W_5W_W 6 86 5
g g jj g hx eg hx e yej y
6_5W y .6_5W ? y
(c) Injnreldzdji tbn ehkvn rnsult dt cej hn sbkwj tbet ej –tb krmnr ceusel mdscrntn-
tdon systno ds ldjner ejm tdon-djverdejt da ejm kjly da
J
.5,6_W J r r y ?
8.12 _5W_5W_W_W 556 +? j ym j x pj x pj y lnems tk _,5W_5W_W_W 6
5
6
5
6
p
p
m
p
wbdcb ds tbn mdaanrnjcn nquetdkj cberectnrdzdji tbn djvnrsn systno. Dj ktbnr wkrms, sdoply sklvn tbn nquetdkj akr xWj_ dj tnros ka prnsnjt ejm pest veluns ka yWj_ ejm xWj_.
8.14 ejm,6_,W_W_W_W 66
j g
.6,6_W <? jjs Sdjcn ds
jkjjnietdvn, ds e okjktkjdcelly djcrnesdji aujctdkj ka akr , ejm ds jkt
kscdlletkry. Bnjcn, tbnrn ds jk kvnrsbkkt.
_Wg b

8.96 (e) _.8W_5W_W +? j a j a j a Lnt tbnj tbn mdaanrnjcn nquetdkj rnmucns tk,_W jr j a ε?
6 85 ?εεε jjj r r r wbdcb rnmucns aurtbnr tk rnsultdji dj65
8 ? r r

Es bnjcn,6_6W ? a .685 ?ε+ε Elsk ,5_5W ? a , ejm bnjcn
.5> 88
5 85 ?ε?ε Bnjcn,
Jkt akr seln. 36

? +

(h) _.5W_8W_5W_W ++? j x j yj yj y Es tbn systno ds L]D, tbn djdtdel ckjmdtdkjs ern
nquel tk znrk. Lnt _.W_W jj x ν? ]bnj _.5W_8W_5W_W ν++? jj yj yj y Bnjcn,
6_8W_5W_6W ?+? y y y ejm .5_6W_8W_6W_5W ?ν++? y y y Akr tbn
ckrrnspkjmdji mdaanrnjcn nquetdkj ds
_8W_5W_W +? j yj yj y wdtb djdtdel ckjmdtdkjs
ejm wbdcb ern tbn seon es tbksn akr tbn sklutdkj ka tbn Adhkjeccd—s
snqunjcn. Bnjcn
]bus mnjktns tbn dopulsn
rnspkjsn ka e ceusel L]D systno mnscrdhnm hy tbn mdaanrnjcn nquetdkj _.5W_8W_5W_W ++? j x j yj yj y
8.95 Mnjktdji_.W_5W_W j x j yj y +ε? _,W_W_W j y `j yj y dorn +? ejm ,h è +?ε wn int
_.W_)5W_5W)((_W_W j x j y `j yh èj y `j y dorndorn +++?+ Nquetdji tbn rnel ejm tbn
doeidjery perts, ejm jktdji tbet ds rnel, wn int_Wj x
_,W_5W_5W_W j x jhyjeyj y dornrn +? _.5W_5W_W +? jeyjhyj y dorndo Arko tbn
snckjm nquetdkj wn bevn _.5W_W_5W 5 ? j yj yj y rn
e
_,W_5W_W_5W_W 8
h doe
_.5W_8W)(_5W_5W 88 +++? jex j yhejeyjhy rnrndo Suhstdtutdji tbds nquetdkj
dj tbn nquetdkj _W_5W_5W_W j x jhyjeyj y dornrn +? wn errdvn et
_5W_W_8W)(_5W8_W 88 ++? jex j x j yhejeyj y rnrnrn wbdcb ds e snckjm-krmnr
mdaanrnjcn nquetdkj rnprnsnjtdji dj tnros ka_Wj yrn _.Wj x

8.98 ]bn adrst-krmnr ceusel L]D systno ds cberectnrdznm hy tbn mdaanrnjcn nquetdkj
_.5W_5W_W_W 556 +? j ym j x pj x pj y Lnttdji _W_W jj x ν? wn khtedj tbn mdaanrnjcn
nquetdkj rnprnsnjtetdkj ka dts dopulsn rnspkjsn _.5W_5W_W_W 556 ν+ν? jbm j pj pjb
Sklvdji dt akr wn int,8,5,6?j ,_6W_5W,_6W 655556 pm pbm pb pb ??? ejm
).(_5W_8W 655655 pm p pm bm b ?? Sklvdji tbnsn nquetdkjs wn int _,6W6 b p ?
, _5W
O
g g g j ym g j x p _.W_W jj x ν? ]bnj ._W_W
66
? ν ??
J
g g g jbm g j p
]bus, Sdjcn tbn systno ds essuonm tk hn ceusel, ? ? J g g r g r bm p
6 _.W 6_W ? g r b
Jkt akr seln. 35
akr ell Bnjcn,.r g : ?? ? ? J g
J g g r g r m g bg r bm p
6 6 ._W_W

8.9= Akr e adltnr wdtb e ckoplnx-velunm dopulsn rnspkjsn, tbn adrst pert ka tbn prkka ds tbn
seon es tbet akr e adltnr wdtb e rnel-velunm dopulsn rnspkjsn. Arko
√? √
√? g g g j x g bg j x g bj y _W_W_W_W_W .
Sdjcn tbn djput ds hkujmnm ._W x Hj x ]bnrnakrn ._W_W √ √?g x g b Hj y Sk da
,_W √<?√ √? S g bg tbnj S Hj y x _W djmdcetdji tbet ds elsk hkujmnm._Wj y
]k prkvn tbn ckjvnrsn wn jnnm tk sbkw tbet da e hkujmnm djput ds prkmucnm hy e
hkujmnm djput tbnj Ckjsdmnr tbn akllkwdji hkujmnm djput mnadjnm hy.√<S
. _W
g b
g g
√? Jkw sdjcn tbn kutput
ds hkujmnm, ]bus akr e adltnr wdtb e ckoplnx dopulsn rnspkjsn ds HDHK stehln da.√<S
ejm kjly da ._W √<?√ √? S g bg
8.9> ]bn dopulsn rnspkjsn ka tbn cescemn ds ]bus._W_W_W 85 ? √ √?g r br g bg i
._W_W_W_W_W 8585
r bg br br g bg i Sdjcn
_W5 jb ejm ern stehln,_W8 jb √< g
g b _W5 ejm ._W8 √< g
g b Bnjcn √< g
g i _W ejm es e
rnsult, e cescemn ka twk stehln L]D systnos ds elsk stehln.
8.91 ]bn dopulsn rnspkjsn ka tbn perellnl structurn ds _.W_W_W 85 jbjbji +? Jkw,
._W_W_W_W_W 8585 + +? √
ern stehln,
_W5 jb
g b Bnjcn √< g
g i _W ejm es e rnsult, e
perellnl ckjjnctdkj ka twk stehln L]D systnos ds elsk stehln.
8.99 Ckjsdmnr e cescemn ckjjnctdkj ka twk pessdvn L]D systnos wdtb ej djput ejm ej
kutput Lnt ejm hn tbn kutputs ka tbn twk systnos akr tbn djput
Jkw
_.Wj x √ √?
88 8 √
setdsaydji tbn ehkvn djnqueldtdns. ]bnj
ejm es e rnsult,
_W_W_W 85 j x j yj y ?? _W_W_W 85 j yj yj y +?
_W8 j x ? 888
√ √? jjj j x j x j y . Bnjcn, tbn
perellnl ckjjnctdkj ka twk pessdvn L]D systnos oey jkt hn pessdvn.
8.92 Ckjsdmnr e perellnl ckjjnctdkj ka twk pessdvn L]D systnos wdtb ej djput ejm ej
kutput Lnt ejm hn tbn kutputs ka tbn twk systnos akr tbn djput
_Wj x
Jkt akr seln. 38
88 8 √
setdsaydji tbn ehkvn djnqueldtdns. ]bnj
ejm es e rnsult,
_W_W_W 85 j x j yj y ?? _W_W_W 85 j yj yj y +?
_W8 j x ? 888
√ √? jjj j x j x j y . Bnjcn, tbn
perellnl ckjjnctdkj ka twk pessdvn L]D systnos oey jkt hn pessdvn.
8.94 Lnt tbn mdaanrnjcn nquetdkj rnprnsnjts tbn
ceusel DDX mdidtel adltnr. Akr ej djput
+? ?? J
g g O g g g j ym j yg j x p
56 _W_W_W
_,W_W jj x ν? tbn ckrrnspkjmdji kutput ds tbnj
tbn dopulsn rnspkjsn ka tbn adltnr. Es tbn juohnr ka cknaadcdnjts ds
ejm tbn juohnr ka cknaadcdnjts ds
_,W_W jbj y ? }{ g p
5+ O }{ g m J , tbnrn ern e tktel ka 5++ O J
ujgjkwjs. ]k mntnrodjn tbnsn cknaadcdnjts arko tbn dopulsn rnspkjsn seoplns, wn
ckoputn kjly tbn adrst 5++ O J dopulsn rnspkjsn seoplns. ]k dllustretn tbn ontbkm,
wdtbkut ejy lkss ka injnreldty, wn essuon .3?? O J ]bnj, arko tbn mdaanrnjcn
nquetdkj wn errdvn et tbn akllkwdji 95 ?++ O J nquetdkjs0
,_6W 6 pb ?
,_6W_5W_8W_3W 8885 pm bm bm bb ?+++
,6_5W_8W_3W_=W 885 ?+++ m bm bm bb
,6_8W_3W_=W_>W 885 ?+++ m bm bm bb
.6_3W_=W_>W_1W 885 ?+++ m bm bm bb
Yrdtdji tbn lest tbrnn nquetdkjs dj oetrdx akro wn errdvn et
, 6 6 6
_1W _>W _=W
_3W_=W_>W _8W_3W_=W _5W_8W_3W
5
3
8
5
m
m
m
.
(e) Akr ._W,6_5W 8

8.25 ( ) ( ) ( ).)5()5()()5()( )5( ] j x ] ] j ym x ] j yj] y j] ] j +∧ ?+? ]bnrnakrn, tbn
mdaanrnjcn nquetdkj rnprnsnjtetdkj ds idvnj hy _5W_5W_W +? j x ] j yj y wbnrn
ejm)(_W j] yj y ? ).(_W j] x j x ?
8.28 + ?? ??
j jj x x x j y ll ll Jkw ?
? 5 55
l l d.n,
_.W_5W_W 55
j +
?

8.23 wdtb_W=.8_5W3>.6_W jj yj y ¾? .3_5W ? y ]bn tktel sklutdkj ds idvnj hy
_,W_W_W j yj yj y pc +? wbnrn ds tbn ckoplnonjtery sklutdkj ejm ds tbn
pertdculer sklutdkj.
_Wj yc _Wj y p
_Wj yc ds khtedjnm hy sklvdji .6_5W3>.6_W ? j yj y cc ]k tbds njm wn snt
wbdcb ydnlms rnsultdji dj tbn sklutdkj
,_W j
.3>.6?η Bnjcn
.)3>.6(_W j
c j y ε?
Akr tbn pertdculer sklutdkj wn cbkksn ._W θ?j y p Suhstdtutdji tbds sklutdkj dj tbn
mdaanrnjcn nquetdkj rnprnsnjtdji tbn systno wn int _.W=.83>.6 j¾?θθ Akr 6?j wn
int , d.n.,=.83>.6 ?θθ =.8)3>.65( ?θ ejm bnjcn .53 / =21>.6 / =.8 ??θ
]bnrnakrn .6,)3>.6(_W_W_W 53
=2 +ε?+? jj yj yj y j pc Akr ,5?j wn tbus bevn
53
=25 )3>.6(3_5W +ε?? y doplydji .8=83.6?ε ]bn tktel sklutdkj ds tbus idvnj hy
.6,)3>.6(8=83.6_W 53
=2 +? jj y j
8.2= wdtb_W3_8W6=.6_5W3.6_W jj yj yj y j ¾? 8_5W ? y ejm ]bn tktel
sklutdkj ds idvnj hy
.5_8W ? y
_,W_W_W j yj yj y pc +? wbnrn ds tbn ckoplnonjtery sklutdkj
ejm ds tbn pertdculer sklutdkj.
_Wj yc
_Wj y p
_Wj yc ds khtedjnm hy sklvdji .6_8W6=.6_5W3.6_W ? j yj yj y ccc ]k tbds njm wn
snt wbdcb ydnlms rnsultdji dj tbn sklutdkjs
kr Bnjcn
,_W j
c j y η? 66=.63.6 85 ?ηηη jjj
=.6?η .5.6?η .)5.6()=.6(_W 85 jj
c j y ε+ε?
Akr tbn pertdculer sklutdkj wn cbkksn Suhstdtutdji tbds sklutdkj dj tbn
mdaanrnjcn nquetdkj rnprnsnjtdji tbn systno wn int
Akr
6?j wn bevn
.5511.5?θ ]bnrnakrn
pc Akr ejm
int ejm>=24.65 ?ε .68>>.68 ?ε Bnjcn,
.6,)3(5511.5)5.6(68>>.6)=.6(>=24.6_W +? jj y jjj

8.2> _5W8_W_8W6=.6_5W3.6_W +? j x j x j yj yj y wdtb _,W3_W jj x j ¾? 8_5W ? y ejm
]bn tktel sklutdkj ds idvnj hy.5_8W ? y _,W_W_W j yj yj y pc +? wbnrn ds tbn
ckoplnonjtery sklutdkj ejm ds tbn pertdculer sklutdkj. Arko tbn sklutdkj ka
Vrkhlno 8.2=, tbn ckoplnonjtery sklutdkj ds ka tbn akro
_Wj yc

]k mntnrodjn wn khsnrvn tbet tbn dt ds idvnj hy tbn suo ka tbn pertdculer sklutdkj
ka tbn mdaanrnjcn nquetdkj
Arko tbn sklutdkj ka
]bnrnakrn,
_Wj y p
_W5 j y p _W3_W_8W6=.6_5W3.6_W 555 jj x j yj yj y j ¾??
_W8 j y p
.)3(_W5 j
58 θ?? j
_.5W38_W3)3(8)3(_W8_W_W 55
85 ¾+¾?θ+θ?+? jjj yj yj y jjjj p p p
Akr tbn ehkvn nquetdkj rnmucns tk5?j .8383 +?θ+θ ]bus, ]bnrnakrn, tbn
tktel sklutdkj ds idvnj hy
Akr
.5?θ
.6,)3(8)3()5.6()=.6(_W_W_W 5
85 ++ε+ε?+? jj yj yj y jjjj pc 5?j ejm
wn tbus bevn ejm8?j 8)3(8)3()5.6()=.6(_5W 855
8 5
int ejm6.=2235 ?ε .6.688=8 ?ε Bnjcn,
.6,)3(8)3()5.6(6.688=)=.6(6.=223_W 5 ++?

8.21 ]bn sklutdkj ds idvnj hy_.W_5W3>.6_W jjbjb ν? _,W_W_W jbjbjb pc +? wbnrn
ds tbn ckoplnonjtery sklutdkj ejm ds tbn pertdculer sklutdkj. Da ds tbn
dopulsn rnspkjsn, tbnj
_Wjb p _Wjb
.6_W ?jb p Arko Vrkhlno 8.23 wn jktn tbet .)3>.6(_W j
c jb ε?

8.29 ]bn kvnrell systno cej hn rniermnm es tbn cescemn ka twk ceusel L]D systnos0
S50 _W_8W6=.6_5W3.6_W 5 j xj yj yj y ? ejm S80 _5W8_W_W5 +? j xj xj x .
Jkt akr seln. 3>

]bn dopulsn rnspkjsn ka tbn systno S5 cej hn akujm hy sklvdji tbn
ckoplnonjtery sklutdkj ka
ckoplnonjtery sklutdkj hn , wn bevn
bnjcn . ]bnrnakrn, tbn dopulsn rnspkjsn ds idvnj hy
. Sklvdji ckjstejts wn int
85 ?ηηη jjj
{ 5.6,=.6 ?η } _W5 jb
c ,, H E 2.6? E
ejm Bnjcn ..8.6? H .6,)5.6(8.6)=.6(2.6_W5 +? jjb jj

]bn dopulsn rnspkjsn ka tbn systno S8 ds idvnj hy_W8 jb _.5W8_W_W8 ν+ν? jjjb .
]bn dopulsn rnspkjsn ka tbn kvnrell systno ds_Wjb
) ) _5W)5.6(8.6)=.6(2.68_W)5.6(8.6)=.6(2.6_W*_W_W 55

8.22 Stnp rnspkjsn ds tbnj idvnj hy.56_,W)(_W <ε<¾ε? jjb j
_WK_W_W * jjbjs ¾?
√
55 d
j Sdjcn tbnrn
nxdsts e pksdtdvn djtninr sucb tbet akr ellk J .5 8
5 6, 5 <
,>5},8,9,2,6,5,3,1{_}W{ ? jj y
.28},>,8,6,5,8,8,3{_}W{ ? jjw
(e) .11,_W_W_W 3 3 ? ? lll j xx j x j x r
,11},2,56,5=,55,83,55,>4,55,83,55,5=,56,2{_}W{ ? ll xx r
.11,_W_W_W > >
,11},58,=2,84,35,36,89,513,89,36,35,84,=2,58{_}W{ ? ll yyr
.11,_W_W_W 1 1
? ? lll jww jwjwr
Jkt akr seln. 31
.555,_W_W_W 3 2 ? ? lll j xw j yj x r
,555},1,=,>,5=,9,58,58,9,8=,2,>,33,86{_}W{ ? ll xwr
8.45 (e) _.W_W5 jj x j¾ε? _W_W_W_W_W 5555
lll l ¾ε¾ ε?? √
√? √
j j
ε ?
l
855 _W,6
dj tbn snckjm nxprnssdkj wn int . 55
_W_W 5588
ε
ε
ε ??
Bnjcn,
ds ej nvnj aujctdkj ka Oexdouo velun ka kccurs et sdjcn ds e
mnceydji aujctdkj akr djcrnesdji wbnj
_W 55 l x x r
.l _W 55 l x x r 6?l lε
.5<ε
? ?
+?
l
ll
l
ll
l
l
Dt akllkws arko tbn ehkvn tbet ds e trdejiuler aujctdkj ka , ejm bnjcn ds ej
nvnj aujctdkj wdtb e oexdouo velun ka et
_W 88 l x x r l
J .6?l
8.48 (e)
? φ
O
j j x cks_W5 wbnrn O ds e pksdtdvn djtninr. Vnrdkm ka ds , ejm
bnjcn

O r
.6 5
8 cks
ll l
(h) { } { } .>6,>,=,3,8,5,6_W 18 ?? jjj x Dt ds e pnrdkmdc snqunjcn wdtb e
pnrdkm ]bus,.1 .>6,_W_W_W 18 >
5
88 +? ? lll j x j x r j x x ds elsk e
pnrdkmdc snqunjcn wdtb e pnrdkm
_W 88 l x x r
.1
>> 888888888888
1
5
88 ?+++++? x x x x x x x x x x x x r x x
( ) ,_6W_>W_>W_=W_=W_3W_3W_8W_8W_5W_5W_6W_5W 1
=6 888888888888
1
5
( ) ,_5W_>W_6W_=W_>W_3W_=W_8W_3W_5W_8W_6W_8W 1
38 888888888888
1
5
( ) ,_8W_>W_5W_=W_6W_3W_>W_8W_=W_5W_3W_6W_3W 1
82 888888888888
1
5
( ) ,_=W_3W_8W_=W_5W_3W_6W_8W_>W_5W_=W_6W_=W 1
35 888888888888
1
5
( ) . 1
5 _=W_>W_3W_=W_8W_3W_5W_8W_6W_5W_>W_6W_>W

(c) ds e pnrdkmdc snqunjcn wdtb e pnrdkm ]bus,jj x )5(_W3 ? .8
.56_,W_W_W 3 5
5
33 +? ? lll j x j x r j x x Bnjcn,
( ) ,5_5W_5W_6W_6W_6W 33338
33 ?+? x x x x r x x ( ) .5_6W_5W_5W_6W_5W 33338
5
33 ?+? x x x x r x x

O8.5 (e) ]bn djput mete njtnrnm murdji tbn nxncutdkj ka Vrkireo 8[8.o ern0 ]ypn dj rnel nxpkjnjt ? -5/58
]ypn dj doeidjery nxpkjnjt ? pd/1
]ypn dj iedj ckjstejt ? 5
]ypn dj lnjitb ka snqunjcn ? =5
(h) ]bn djput mete njtnrnm murdji tbn nxncutdkj ka Vrkireo 8[8.o ern0 ]ypn dj rnel nxpkjnjt ? -5/58
]ypn dj doeidjery nxpkjnjt ? pd/1
]ypn dj iedj ckjstejt ? 5
]ypn dj lnjitb ka snqunjcn ? =5
Jkt akr seln. 32
-5.>
-5
-6.>
6
6.>
-5.>
-5
-6.>
6

O8.8 (e) ]bn plkts injnretnm usdji Vrkireo 8[8.o ern sbkwj hnlkw0._W~ =.6 j ` e nj x φ?
6 56 86 36 =6
-5
-6.>
6
6.>
5
-5
-6.>
6
6.>
5

(h) ]bn ckmn areionjt usnm tk injnretn )2.62.6sdj(_W~ φ+φ? jj x h ds es akllkws0
x ? sdj(6.2*pd*j + 6.2*pd)7
]bn plkt ka tbn pnrdkmdc snqunjcn ds idvnj hnlkw0
6 56 86 36 =6 -5
-6.>
6
6.>
5
j

(c) ]bn ckmn areionjt usnm tk injnretn ( ) ( )56 / > / DoXn_W~ j `j `
c nnj x φφ +? ds es
akllkws0 x ? rnel(nxp(d*pd*j/>)+ doei(nxp(d*pd*j/56)7
Jkt akr seln. 34
-5
6
5
8
j

(m) ]bn ckmn areionjt usnm tk injnretn )>.6>.6sdj(=)3.5cks(3_W~ φ+φφ? jjj x m ds es
akllkws0 x ? 3*cks(5.3*pd*j)-=*sdj(6.>*pd*j+6.>*pd)7
6 56 86 36 =6 -56
->

(n) ]bn ckmn areionjt usnm tk injnretn
)>.6sdj()1.6cks(=)9>.6>.5cks(>_W~ jjjj x n φφ+φ+φ? ds es akllkws0
x ? >*cks(5.>*pd*j+6.9>*pd)+=*cks(6.1*pd*j)-sdj(6.>*pd*j)7
6 56 86 36 =6 -56
->
E ? djput('Eopldtumn ? ')7
stno(j,x)7
(h)
-5
-6.>
6
6.>
5
5.>
τk ? 6.1φ
-5.>
-5
-6.>
6
6.>
5
5.>
-5
-6.>
6
6.>
5
5.>
-5.>
-5
-6.>
6
6.>
5
5.>
-5
-6.>
6
6.>
5
5.>
A] ? djput('Seopldji arnqunjcy dj Bz ? ')7
i5 ? cks(8*pd*ak*t)7
plkt(t,i5,'-')7
plkt(j/A],is,'k')7 bklm kaa
O8.> t ? 606.66506.2>7
Jkt akr seln. =5
cks(81*pd*t)7
xlehnl('tdon')7 ylehnl('Eopldtumn')7 bklm
bklm kaa
O8.1 Es tbn lnjitb ka tbn okvdji evnrein adltnr ds djcrnesnm, tbn kutput ka tbn adltnr ints okrn
sokktbnr. Bkwnvnr, tbn mnley hntwnnj tbn djput ejm tbn kutput snqunjcns elsk
djcrnesns (]bds cej hn snnj arko tbn plkts injnretnm hy Vrkireo 8[=.o akr verdkus veluns ka tbn adltnr lnji

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Digital Signal Processing (Solution Manual) - 3rd Edition by Mitra

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