VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF ENGINEERING AND TECHNOLOGY
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E#AMINATION $UESTIONS AND E#ERCISES
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9uestion 28 *i)* one o( t*e (ollo;ing state!ents is INCORRECT< A8 An energ energ signal signal )an not not 'e 'e .eriodi)8 .eriodi)8 =8 A .o;er .o;er signal signal )an )an not *a4e (inite (inite energ8 energ8 C8 A sinuso sinusoida idall is a .o; .o;er er signal signal88 "8 A (inite-len (inite-lengt* gt* signal signal )an )an 'e a .o;er .o;er signal8 signal8 9uestion #8 *i)* one o( t*e (ollo;ing state!ents is INCORRECT< A8 T*e i!.ulse i!.ulse res.onse res.onse o( a )ausal )ausal LTI LTI sste! is is a )ausal signal8 signal8 =8 T*e i!.ulse i!.ulse res.onse res.onse o( a sta'le sta'le LTI LTI sste! is is an energ energ signal8 signal8 C8 T*e i!.ulse i!.ulse res.onse res.onse o( a sta'le sta'le LTI LTI sste! is is a (inite-lengt* (inite-lengt* signal signal88 "8 T*e (re1uen (re1uen) ) res.onse res.onse o( a dis)ret dis)rete-ti e-ti!e !e sta'le sta'le LTI LTI sste! sste! is a )ontinuou )ontinuouss(re1uen) (un)tion8 E8 T*e (re1uen) (re1uen) res.onse res.onse o( o( a dis)rete-ti! dis)rete-ti!ee LTI sste! sste! is .eriodi .eriodi)8 )8 9uestion %8 >i4en a sste! des)ri'ed ' t*e e1uation y e1uation y++n/− y+ y+n−2/ ? x ? x++n/, ;*i)* one o( t*e (ollo;ing state!ents is INCORRECT a'out t*is sste!< A8 T*e T*e sste sste! ! is lin linea ear8 r8 =8 T*e sste! sste! is is ti!e-in ti!e-in4ar 4arian iant8 t8 C8 T*e T*e sste sste! ! is sta sta'l 'le8 e8 "8 T*e T*e sste sste! ! is )aus )ausal al88 Types of credit hours: Lecture hours; Tutorial/Lab hours; Self preparatory hours.
9uestion 08 *i)* one o( t*e (ollo;ing state!ents is INCORRECT< A8 All .oles o( a sta'le sta'le )ontinuous-t )ontinuous-ti!e i!e )ausal LTI LTI sste! !ust !ust 'e in t*e rig*t *al( *al( o( t*e s-.lane8 =8 All .oles .oles o( a sta'le dis)rete-ti! dis)rete-ti!ee )ausal LTI sste! sste! !ust !ust 'e inside t*e t*e unit )ir)le )ir)le in t*e @-.lane8 C8 T*e region region o( )on4ergen)e )on4ergen)e +ROC/ +ROC/ o( t*e trans(er trans(er (un)tion (un)tion o( a sta'le )ontinuo )ontinuoususti!e LTI sste! !ust )ontain t*e j t*e jω ais ais o( t*e s-.lane8 "8 T*e region region o( )on4er )on4ergen gen)e )e +ROC/ o( t*e trans(er trans(er (un)tio (un)tion n o( a sta'le sta'le dis)retedis)reteti!e LTI sste! !ust )ontain t*e unit )ir)le in t*e @-.lane8 9uestion &8 *i)* one o( t*e (ollo;ing signals is NOT .eriodi): #
A8
x ( t )=[ )os ( # π t )]
=8
( t ≤− ≤−2∨t > 2) x ( t )= ∑ (t −% k ) , ;*ere ( t )= )= 2+ t (−2 < t ≤ $ ) k =−∞ 2 −t ( $ < t ≤2 )
C8
( t ≤− ≤−2 ∨t > 2 ) x ( t )= ∑ (t − # k ) , ;*ere ( t )= )= 2 + t (−2 < t ≤ $ ) k =−& ( $ < t ≤2 ) 2 −t
"8
x ( n )=(−2 )
{ {
$
+∞
&
$
n
9uestion B8 *i)* one o( t*e (ollo;ing signals is NOT .eriodi): A8 x ( n)= )os ( # n ) =8
x ( n)= )os( # π n)
C8
x ( n)=
+∞
{(−2 ) [ δ ( n− # k )+δ(n + % k )]} )] } ∑ =−∞ =−∞ k
k
"8
x ( n )=#sin ( 0 π n / 23 )+ )os ( 2$ π n / 23 )+ 2
9uestio 9uestion n 68 *i)* *i)* o( t*e sste! sste!ss des)ri' des)ri'ed ed ' t*e (ollo; (ollo;ing ing i!.ulse i!.ulse res.on res.onses ses is !e!orless< A8
)=)os (π t ) h ( t )=
=8
)= e h ( t )=
C8
)= u (t + 2 ) h ( t )=
"8
)=% δ( t ) h ( t )=
−# t
u ( t − 2)
9uestio 9uestion n 8 *i)* *i)* o( t*e sste! sste!ss des)ri' des)ri'ed ed ' t*e (ollo; (ollo;ing ing i!.ulse i!.ulse res.on res.onses ses is !e!orless< n
A8
h ( n )=(−2 ) u (−n )
=8
h ( n )=(2 / # )
C8
h ( n )= # [ u ( n )−u ( n −2 )]
"8
h ( n )=)os ( π n / C)[ u ( n )−u ( n −2$ )]
∣n∣
n
9uestio 9uestion n 38 *i)* *i)* o( t*e sste! sste!ss des)ri' des)ri'ed ed ' t*e (ollo; (ollo;ing ing i!.ulse i!.ulse res.on res.onses ses is )ausal< #
A8
)=)os (π t ) h ( t )=
=8
)= e h ( t )=
C8
)= u (t + 2 ) h ( t )=
"8
h ( t )=)os ( π t ) u (−t )
−# t
u ( t − 2)
9uestion 2$8 *i)* o( t*e sste!s des)ri'ed ' t*e (ollo;ing i!.ulse res.onses is )ausal< n
A8
h ( n )=(−2 ) u (−n )
=8
h ( n )=(2 / # )
C8
h ( n )=u ( n )−# u ( n− & )
"8
h ( n )=sin (π n / # )
∣n∣
9uestion 228 *i)* o( t*e sste!s des)ri'ed ' t*e (ollo;ing i!.ulse res.onses is sta'le< A8
)=)os (π t ) h ( t )=
=8
)= e h ( t )=
C8
)= u (t + 2 ) h ( t )=
"8
h ( t )=)os ( π t ) u (−t )
−# t
u ( t − 2)
9uestion 2#8 *i)* o( t*e sste!s des)ri'ed ' t*e (ollo;ing i!.ulse res.onses is sta'le< A8
h ( n )=sin ( π n / # )
=8
h ( n )=(−2 ) u (− n)
C8
h ( n )=u ( n )−# u ( n− & )
"8
n
+∞
∑= δ ( n− # k )
h ( n )=
k $
9uestion 2%8 *i)* o( t*e (ollo;ing sste!s is a )ausal linear ti!e-in4ariant sste!< A8
y ( t )=(t −2 ) x ( t )
=8
= x t − # x t / # y t =
C8
y n = x n y n−2
"8
=∣ x n − x n −2 ∣ y n =∣
9uestion 208 *at is t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal: −t
x ( t )= e )os ( # π t ) u ( t ) A8 T*e )ontinu )ontinuous-ti ous-ti!e !e 5ourier 5ourier trans(o trans(or! r! +5T/8 +5T/8 =8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier trans(or trans(or! ! +"T5T/ +"T5T/88 C8 T*e )ontinuou )ontinuous-ti!e s-ti!e 5ourier 5ourier series series +5S/8 "8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier series series +"T5S/8 +"T5S/8 %
9uestion 2&8 *at is t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal:
{
x ( n)=
)os ( π n / 2$ )+ j sin ( π n / 2$ ) $
(∣n∣<2$ ) (∣n∣≥2$ )
A8 T*e )ontinu )ontinuous-ti ous-ti!e !e 5ourier 5ourier trans(o trans(or! r! +5T/8 +5T/8 =8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier trans(or trans(or! ! +"T5T/ +"T5T/88 C8 T*e )ontinuou )ontinuous-ti!e s-ti!e 5ourier 5ourier series series +5S/8 "8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier series series +"T5S/8 +"T5S/8 9uestion 2B8 *at is t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal: 2 +t
x ( t )= e u ( # −t ) A8 T*e )ontinu )ontinuous-ti ous-ti!e !e 5ourier 5ourier trans(o trans(or! r! +5T/8 +5T/8 =8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier trans(or trans(or! ! +"T5T/ +"T5T/88 C8 T*e )ontinuou )ontinuous-ti!e s-ti!e 5ourier 5ourier series series +5S/8 "8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier series series +"T5S/8 +"T5S/8 9uestion 268 *at is t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal: )=∣sin ( # π t )∣ x ( t )=∣ A8 T*e )ontinu )ontinuous-ti ous-ti!e !e 5ourier 5ourier trans(o trans(or! r! +5T/8 +5T/8 =8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier trans(or trans(or! ! +"T5T/ +"T5T/88 C8 T*e )ontinuou )ontinuous-ti!e s-ti!e 5ourier 5ourier series series +5S/8 "8 T*e dis)rete-ti!e dis)rete-ti!e 5ourier 5ourier series series +"T5S/8 +"T5S/8 9uestion 28 *at is t*e a..ro.riate des)ri.tion o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse: − # t
h ( t )=δ( t )− # e
u ( t )
A8 A lo;lo;-.a .ass ss (ilt (ilter er88 =8 A *ig *ig**-.a .ass ss (il (ilte ter8 r8 C8 A 'an 'andd-.a .ass ss (il (ilte ter8 r8 "8 A 'and-r 'and-reDe) eDe)tt (ilt (ilter8 er8 9uestion 238 *at is t*e a..ro.riate des)ri.tion o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse: −# t
h ( t )= 0 e
)os ( &$ t )
A8 A lo;lo;-.a .ass ss (ilt (ilter er88 =8 A *ig *ig**-.a .ass ss (il (ilte ter8 r8 C8 A 'an 'andd-.a .ass ss (il (ilte ter8 r8 "8 A 'and-r 'and-reDe) eDe)tt (ilt (ilter8 er8 9uestion #$8 *at is t*e a..ro.riate des)ri.tion o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse:
()
2 6 h ( n )= C C 0
n
u(n)
A8 A lo;lo;-.a .ass ss (ilt (ilter er88 =8 A *ig *ig**-.a .ass ss (il (ilte ter8 r8 C8 A 'an 'andd-.a .ass ss (il (ilte ter8 r8 "8 A 'and-r 'and-reDe) eDe)tt (ilt (ilter8 er8 9uestion #28 *at is t*e a..ro.riate des)ri.tion o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse:
{
(−2 )n (∣n∣≤2$ ) h ( n )= (∣n∣> 2$ ) $ A8 A lo;lo;-.a .ass ss (ilt (ilter er88 =8 A *ig *ig**-.a .ass ss (il (ilte ter8 r8 C8 A 'an 'andd-.a .ass ss (il (ilte ter8 r8 "8 A 'and-r 'and-reDe) eDe)tt (ilt (ilter8 er8 9uestion ##8 *at is t*e initial 4alue o( t*e signal x ( t ) , gi4en its La.la)e trans(or! as (ollo;s: 2 ! ( s )= # s + & s −# A8 $ =8 2 C8 # "8
2
−
9uestion #%8 *at is t*e initial 4alue o( t*e signal x ( t ) , gi4en its La.la)e trans(or! as (ollo;s: s +# ! ( s )= # s + # s − % A8 $ =8 2 C8 # "8
2
−
9uestion #08 *at is t*e initial 4alue o( t*e signal x ( t ) , gi4en its La.la)e trans(or! as (ollo;s: −# s
! ( s )=e A8 $ =8 2 C8 # "8
2
−
&
#
+ s s + # s− # B s
#
9uestion #&8 *at is t*e (inal 4alue o( t*e signal x ( t ) , gi4en its La.la)e trans(or! as (ollo;s: #
+% ! ( s )= # s + & s + 2 # s
A8 $ =8 # C8 2# "8 20 9uestion #B8 *at is t*e (inal 4alue o( t*e signal x ( t ) , gi4en its La.la)e trans(or! as (ollo;s: s +# ! ( s )= % # s + # s + s A8 $ =8 # C8 2# "8 20 9uestion #68 *at is t*e (inal 4alue o( t*e signal x ( t ) , gi4en its La.la)e trans(or! as (ollo;s: ! ( s )=e
−% s
#
+2 # s ( s + # ) # s
A8 $ =8 # C8 2# "8 20 9uestion #8 *i)* one o( t*e sste!s des)ri'ed ' t*e (ollo;ing trans(er (un)tions )an NOT 'e 'ot* )ausal and sta'le<
( s + 2 )( s + #) ( s + 2 )( s + # s + 2$ )
A8
" ( s )=
=8
s − % s + # " ( s )= ( s + #)( s #− # s + )
C8
" ( s )=
+ # s −% ( s + % )( s # + # s +& )
"8
( s + 2 )( s + # s + 2$ ) ( s )= " ( ( s + 2 )( s + #)
#
#
s
#
#
9uestion #38 5or ;*i)* o( t*e (ollo;ing signals does t*e dis)rete-ti!e 5ourier trans(or! NOT eist< A8
x ( n)=δ( n−2 ) B
=8
x ( n)=δ( n + 2 )
C8
x ( n)=( # / % )
"8
x ( n)=( 2 / 0 ) u (−n )
∣n∣ n
9uestion %$8 *i)* one o( t*e sste!s des)ri'ed ' t*e (ollo;ing trans(er (un)tions )an 'e 'ot* )ausal and sta'le< A8
# # + % " ( # )= # # + # −& / 2B −2
=8
" ( # )=
#
[ 2−( 2 / # ) # −2 ]( 2 + %@−2) #
# −2 / 0
C8
" ( # )=
"8
# " ( # )= −2 −# 2−( 2 / # ) # +( 2 / 0) #
#
B # + 6 # + % −#
,&*&
E-erc!e!
Eer)ise 28 5ind t*e (unda!ental .eriod o( t*e (ollo;ing .eriodi) signal: #
x ( t )=[ )os ( # π t ) ]
Eer)ise #8 5ind t*e (unda!ental .eriod o( t*e (ollo;ing .eriodi) signal:
{
(t ≤−2∨t > 2 ) x ( t )= ∑ (t −% k ) , ;*ere ( t )= 2 + t (−2 < t ≤ $) k =−∞ ( $
$
Eer)ise %8 5ind t*e (unda!ental .eriod o( t*e (ollo;ing .eriodi) signal: n
x ( n )=(−2 )
Eer)ise 08 5ind t*e (unda!ental .eriod o( t*e (ollo;ing .eriodi) signal: x ( n )=)os ( # π n ) Eer)ise Eer)ise &8 "eter!ine "eter!ine t*e i!.ulse i!.ulse res.onse res.onse o( a )ontinuous-ti!e )ontinuous-ti!e LTI sste! des)ri'ed des)ri'ed ' t*e (ollo;ing e1uation: #
dy (t ) d y ( t ) dx ( t ) +# = y ( t )+ % # dt dt dt Eer)ise B8 "eter!ine t*e !agnitude res.onse and t*e .*ase res.onse o( a )ontinuousti!e LTI sste! de)sri'ed ' t*e (ollo;ing e1uation: #
dy (t ) d y ( t ) dx ( t ) +# = y ( t )+ % # dt dt dt Eer Eer)is )isee 68 "ete "eter! r!in inee t*e t*e trans trans(e (err (un) (un)ti tion on o( a )aus )ausal al dis)r dis)ret ete-t e-ti! i!ee LTI LTI sste sste! ! des)ri'ed ' t*e (ollo;ing e1uation: y ( n )+ % y ( n− 2)+ # y ( n− # )= x ( n −2 ) Eer)ise 8 "eter!ine t*e ste. res.onse o( a )ausal dis)rete-ti!e LTI sste! des)ri'ed ' t*e (ollo;ing e1uation: 6
y ( n )+ % y ( n− 2)+ # y ( n− # )= x ( n −2 ) Eer)ise 38 A dis)rete-ti!e LTI sste! *as t*e (ollo;ing i!.ulse res.onse: −n
h ( n )= # u ( n ) "eter!ine t*e out.ut o( t*e sste! ;*en t*e in.ut signal is: x ( n )=sin ( π n / % + π / 0 )+ # Eer)ise 2$8 "eter!ine t*e i!.ulse res.onse o( t*e (ollo;ing )ausal LTI sste!: dy t − y t = x t x t −2 dt Eer)ise 228 "eter!ine t*e i!.ulse res.onse o( t*e (ollo;ing )ausal LTI sste!: 0 y n − y n −# = x n − # x n − 2 Eer)ise 2#8 T*e negati4e (eed'a) )ontrol sste! s*o;n in (igure 'ello; *as a .lant P and a (eed'a) )oe((i)ient o( $ o( $ , in ;*i)* t*e .lant P is des)ri'ed ' t*e e1uation dy ( t ) − # y ( t )= x ( t ) dt and $ and $ is is a real 4alue8
x+t /
−
P
y+t /
$ Co!.ute t*e trans(er (un)tion o( t*e (eed'a) )ontrol sste!8 Eer)ise 2%8 T*e negati4e (eed'a) )ontrol sste! s*o;n in (igure 'ello; *as a .lant P and a (eed'a) )oe((i)ient o( $ o( $ , in ;*i)* t*e .lant P is des)ri'ed ' t*e e1uation dy ( t ) − # y ( t )= x ( t ) dt and $ and $ is is a real 4alue8
x+ x+t /
−
P
y+t /
$ "eter!ine $ "eter!ine $ so so t*at t*e sste! is )ausal and sta'le8 Eer)ise Eer)ise 208 "eter!ine "eter!ine t*e (re1uen) (re1uen) res.onse o( an LTI sste! *a4ing *a4ing t*e (ollo;ing (ollo;ing i!.ulse res.onse:
)=sin ( t )[ )[ u ( t )−u ( t −2 )] h ( t )= Eer)i Eer)ise se 2&8 "eter!ine "eter!ine t*e res.on res.onse se o( an LTI sste! *a4ing *a4ing t*e (ollo;in (ollo;ing g i!.uls i!.ulsee res.onse:
)=sin ( t )[ )[ u ( t )−u ( t −2 )] h ( t )= to t*e in.ut signal:
x ( t )=)os ( # t + π / 0 )+ 2
Eer)ise 2B8 "eter!ine t*e ste. res.onse o( t*e sste! *a4ing t*e (ollo;ing i!.ulse res.onse: n
h ( n )=(−2 / # ) u ( n ) Eer)ise 268 "eter!ine t*e ste. res.onse o( t*e sste! *a4ing t*e (ollo;ing i!.ulse res.onse: h ( n )=δ( n )−δ( n− # ) Eer)ise 28 "eter!ine t*e ste. res.onse o( t*e sste! *a4ing t*e (ollo;ing i!.ulse res.onse: n
h ( n )=(−2 ) [ u ( n + # )− u ( n− % )] Eer)ise 238 "eter!ine t*e ste. res.onse o( t*e sste! *a4ing t*e (ollo;ing i!.ulse res.onse: h ( n )= nu ( n ) Eer)ise #$8 "eter!ine t*e ste. res.onse o( t*e sste! *a4ing t*e (ollo;ing i!.ulse res.onse:
)=(2 / 0 )[ u ( t )−u ( t − 0 )] h ( t )=( Eer)ise #28 "eter!ine t*e ste. res.onse o( t*e sste! *a4ing t*e (ollo;ing i!.ulse res.onse:
)= u (t ) h ( t )= Eer)i Eer)ise se ##8 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((erential e1uation: dy ( t ) + 2$ y )= # x (t ) 2$ y ( t )= dt Eer)i Eer)ise se #%8 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e &
(ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dy ( t ) dx ( t ) + y ( t )= dt dt
+B
Eer)i Eer)ise se #08 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dx ( t ) dt
+ 0 y ( t )= )=%
Eer)i Eer)ise se #&8 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dy (t ) + # y ( t )= x (t ) dt
+#
Eer)i Eer)ise se #B8 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dy (t ) dx (t ) + y (t )= dt dt
+#
Eer)i Eer)ise se #68 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((eren)e e1uation:
y ( n )−ay ( n −2 )=# x ( n ) Eer)i Eer)ise se #8 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n− 2 ) 3
Eer)i Eer)ise se #38 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )+( 3 / 2B ) y ( n − # )= x ( n− 2) Eer)i Eer)ise se %$8 "eter!ine "eter!ine t*e *o!oge *o!ogeneo neous us soluti solution on (or t*e sste! sste! des)ri' des)ri'ed ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )+ y ( n −2 )+( 2 / 0 ) y ( n − # )= x ( n )+ # x ( n −2 ) Eer)ise %28 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: &
dy ( t ) + 2$ y )= # x (t ) 2$ y ( t )= dt
gi4en t*e in.ut: x ( t )=# Eer)ise %#8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: &
dy ( t ) + 2$ y )= # x (t ) 2$ y ( t )= dt
gi4en t*e in.ut: x ( t )=e
−t
Eer)ise %%8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: &
dy ( t ) + 2$ y )= # x (t ) 2$ y ( t )= dt
gi4en t*e in.ut: x ( t )=)os ( % t ) Eer)ise %08 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dx ( t ) dt
+ 0 y ( t )= )=%
gi4en t*e in.ut: x ( t )=t Eer)ise %&8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dx ( t ) dt
+ 0 y ( t )= )=%
gi4en t*e in.ut: x ( t )=e
−t
Eer)ise %B8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dx ( t ) dt
+ 0 y ( t )= )=%
gi4en t*e in.ut: x ( t )=)os ( t )+ sin ( t ) Eer)ise %68 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: 2$
#
d y ( t ) #
dt
dy (t ) dx (t ) + y (t )= dt dt
+#
gi4en t*e in.ut: x ( t )=e
−% t
Eer)ise %8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt gi4en t*e in.ut x ( t )=# e
−t
dy (t ) dx (t ) + y (t )= dt dt
+#
8
Eer)ise %38 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dy (t ) dx (t ) + y (t )= dt dt
+#
gi4en t*e in.ut:
x ( t )=#sin ( t ) Eer)ise 0$8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
y ( n )−( # / & ) y ( n− 2)= # x ( n ) gi4en t*e in.ut: x ( n )=# u ( n ) Eer)ise 028 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
y ( n )−( # / & ) y ( n− 2)= # x ( n ) gi4en t*e in.ut: n
x ( n )=−( 2 / # ) u ( n ) Eer)ise 0#8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
y ( n )−( # / & ) y ( n− 2)= # x ( n ) gi4en t*e in.ut: x ( n )=)os (π n / & ) Eer)ise 0%8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n− 2 ) gi4en t*e in.ut:
x ( n )=nu ( n ) Eer)ise 008 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n− 2 ) gi4en t*e in.ut: n
x ( n )=(2 / ) u ( n ) Eer)ise 0&8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n− 2 ) gi4en t*e in.ut: 22
j π n / 0
x ( n )=e u ( n) Eer)ise 0B8 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n− 2 ) gi4en t*e in.ut: n
x ( n )=(2 / # ) u ( n ) Eer)ise 068 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
y ( n )+ y ( n −2 )+( 2 / # ) y ( n − # )= x ( n )+ #A ( n −2 ) gi4en t*e in.ut: x ( n )=u ( n ) Eer)ise 08 "eter!ine t*e .arti)ular solution (or t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )+ y ( n −2 )+( 2 / # ) y ( n − # )= x ( n )+ # ( n −2 ) gi4en t*e in.ut: n
x ( n )=(−2 / # ) u ( n ) Eer)ise 038 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: dy ( t ) + 2$ y )= # x (t ) 2$ y ( t )= dt gi4en t*e in.ut:
x ( t )=u ( t ) and t*e initial )ondition: y ( $ − )=2 Eer)ise &$8 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dy (t ) dx (t ) + y (t )= dt dt
+#
gi4en t*e in.ut:
x ( t )=sin ( t ) u ( t ) and t*e initial )onditions:
∣
dy ( t ) dt
y ( $ − )=$ and
=2
t = $ −
Eer)ise &28 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
+B
dy ( t ) + y ( t )=# x ( t ) dt
gi4en t*e in.ut: − t
x ( t )=e u ( t ) and t*e initial )onditions:
y ( $ −)=−2 and
∣
dy ( t ) dt
=2
t =$−
Eer)ise "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: 2#
#
d y ( t ) dx ( t ) % + ( )= ) = y t # dt dt gi4en t*e in.ut: − t
x ( t )=# te u ( t ) and t*e initial )onditions:
∣
dy ( t ) dt
y ( $ −)=−2 and
=2
t =$−
Eer)ise &%8 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: dy ( t ) + 2$ y )= # x (t ) 2$ y ( t )= dt gi4en t*e in.ut:
x ( t )=u ( t ) and t*e initial )ondition: y ( $ − )=2 Eer)ise &08 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
dy (t ) dx (t ) + y (t )= dt dt
+#
gi4en t*e in.ut:
x ( t )=sin ( t ) u ( t ) and t*e initial )onditions:
∣
dy ( t ) dt
y ( $ − )=$ and
=2
t = $ −
Eer)ise &&8 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
+B
dy ( t ) + y ( t )=# x ( t ) dt
gi4en t*e in.ut: − t
x ( t )=e u ( t ) and t*e initial )onditions:
y ( $ −)=−2 and
∣
dy ( t ) dt
=2
t =$−
Eer)ise &B8 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
+ y ( t )= )=%
dx ( t ) dt
gi4en t*e in.ut: − t
x ( t )=# te u ( t ) and t*e initial )onditions: y ( $ −)=−2 and
∣
dy ( t ) =2 dt t =$−
Eer)ise &68 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: 2%
y ( n )−(2 / # ) y ( n−2 )= # x ( n ) gi4en t*e in.ut: n
x ( n )=(−2 / # ) u ( n ) and t*e initial )ondition: y (−2)=% Eer)ise &8 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 3 ) y ( n − # )= x ( n −2 ) gi4en t*e in.ut:
x ( n )=u ( n ) t*e initial )onditions: y (−2)=2 and y (−# )=$ Eer)ise &38 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
y ( n )+( 2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n −2 ) gi4en t*e in.ut n
x ( n )=(−2 ) u ( n ) and t*e initial )onditions: y (−2)= 0 and y (−# )=−# Eer)ise B$8 "eter!ine t*e out.ut o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−( % / 0 ) y ( n −2 )+( 2 / ) y ( n −# )= # x ( n ) gi4en t*e in.ut:
x ( n )=# u ( n ) and t*e initial )onditions: y (−2)=2 and y (−# )=−2 Eer)ise B28 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / # ) y ( n−2 )= # x ( n ) gi4en t*e in.ut: n
x ( n )=(−2 / # ) u ( n ) and t*e initial )ondition: y (−2)=% Eer)ise B#8 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−(2 / 3 ) y ( n − # )= x ( n −2 ) gi4en t*e in.ut:
x ( n )=u ( n ) and t*e initial )onditions: y (−2)=2 and y (−# )=$ Eer)ise B%8 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )+( 2 / 0 ) y ( n− 2 )−(2 / C ) y ( n− # )= x ( n )+ x ( n −2 ) gi4en t*e in.ut: 20
n
x ( n )=(−2 ) u ( n ) and t*e initial )onditions:
y (−2)= 0 and y (−# )=−# Eer)ise B08 "eter!ine t*e natural and (or)ed res.onses o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation: y ( n )−( % / 0 ) y ( n −2 )+( 2 / C ) y ( n −# )= # x ( n ) gi4en t*e in.ut: x ( n )=# u ( n ) and t*e initial )onditions:
y (−2)=2 and y (−# )=−2 Eer)ise B&8 5ind t*e di((eren)e e1uation des)ri'ing t*e dis)rete-ti!e sste! re.resented ' t*e (ollo;ing 'lo) diagra!: diagra!:
#
! + s/ s/
s
−
% + s/ s/
+
# Eer)ise BB8 5ind t*e di((eren)e e1uation des)ri'ing t*e dis)rete-ti!e sste! re.resented ' t*e (ollo;ing 'lo) diagra!: diagra!:
s
! + s/ s/
s
−
% + s/ s/
Eer)i Eer)ise se B68 "ra; t*e 'lo)-d 'lo)-diag iagra! ra! re.rese re.resenta ntatio tion n o( a dis)ret dis)rete-ti e-ti!e !e LTI sste! sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
[/
A =
2
−2 / #
2 %
$
]
,
B
[]
=2
,
#
C =
[2
2 ] , and
D=
[ $]
Eer)i Eer)ise se B8 "ra; t*e 'lo)-d 'lo)-diag iagra! ra! re.rese re.resenta ntatio tion n o( a dis)ret dis)rete-ti e-ti!e !e LTI sste! sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
[/
2 2 %
A =
−2 / # $
]
[] 2 #
B=
,
C =
,
[ 2 −2 ]
, and
D=
[ $]
Eer)i Eer)ise se B38 "ra; t*e 'lo)-d 'lo)-diag iagra! ra! re.rese re.resenta ntatio tion n o( a dis)ret dis)rete-ti e-ti!e !e LTI sste! sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
[
A =
$ 2/ %
]
−2 / # −2
,
B
[]
=
$ 2
,
C =
[2
$ ] , and
D=
[ 2]
Eer)i Eer)ise se 6$8 "ra; t*e 'lo)-d 'lo)-diag iagra! ra! re.rese re.resenta ntatio tion n o( a dis)ret dis)rete-ti e-ti!e !e LTI sste! sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
2&
A
[ ]
=$ $ $
[]
B=
,
2
# %
C =
,
[ 2 −2 ]
, and
D=
[ $]
Eer)ise 628 "ra; t*e 'lo)-diagra! re.resentation o( a )ontinuous-ti!e LTI sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
[/
A =
2 % $
$ −2 / #
]
[− ] 2 #
B=
,
,
C =
[2
2 ] , and
D=
[ $]
Eer)ise 6#8 "ra; t*e 'lo)-diagra! re.resentation o( a )ontinuous-ti!e LTI sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
[ ] 2 2 2 $
A =
[ ]
B=
,
−2 #
C =
,
[ $ −2 ]
D=
, and
[ $]
Eer)ise 6%8 "ra; t*e 'lo)-diagra! re.resentation o( a )ontinuous-ti!e LTI sste! des)ri'ed ' t*e (ollo;ing state-4aria'le re.resentation !atri)es:
[
A =
2 2
−# 2
]
,
[]
B=
# %
,
C =
[2
2 ] , and
D=
[ $]
Eer)ise Eer)ise 608 "eter!ine t*e dis)rete-ti!e dis)rete-ti!e 5ourier series re.resentation re.resentation (or t*e (ollo;ing signal: x ( n )=)os ( B π n / 26 +π / % ) 8 Eer)ise Eer)ise 6&8 "eter!ine t*e dis)rete-ti!e dis)rete-ti!e 5ourier series re.resentation re.resentation (or t*e (ollo;ing signal:
x ( n)= #sin ( 0 π n / 23)+ )os ( 2$ π n / 23 )+ 2 8 Eer)ise Eer)ise 6B8 "eter!ine t*e dis)rete-ti!e dis)rete-ti!e 5ourier series re.resentation re.resentation (or t*e (ollo;ing signal: x ( n)=
+∞
{(−2 ) [ δ ( n− # k )+δ( n + % k )]} )] } ∑ =−∞ =−∞ k
8
k
Eer)ise 668 "eter!ine t*e ti!e-do!ain signal re.resented ' t*e (ollo;ing dis)reteti!e 5ourier series )oe((i)ients: ! k = )os ( π k / #2 ) Eer)ise 68 "eter!ine t*e ti!e-do!ain signal re.resented ' t*e (ollo;ing dis)reteti!e 5ourier series )oe((i)ients: ! k = )os ( 2$ π k / 23 )+ # j sin ( 0 π k / 23 ) Eer)ise 638 "eter!ine t*e ti!e-do!ain signal re.resented ' t*e (ollo;ing dis)reteti!e 5ourier series )oe((i)ients: ! k =
+∞
{(−2) ∑ =−∞
&
[ δ ( k − # &)− # δ ( k + % & )]} )] }
&
Eer)ise $8 "eter!ine t*e 5ourier series re.resentation (or t*e (ollo;ing signal:
x ( t )=sin ( % π t )+ )os ( 0 π t ) 8 Eer)ise 28 "eter!ine t*e 5ourier series re.resentation (or t*e (ollo;ing signal: x ( t )=
+∞
{(− {(−2 ) [ δ ( t − k / %)+δ( t + # k / % )]} )] } ∑ =−∞ =−∞ k
8
k
Eer)ise #8 "eter!ine t*e 5ourier series re.resentation (or t*e (ollo;ing signal: x ( t )=
+∞
[ e π / δ (t − # k )] ∑ =−∞ # k 6
8
k
Eer)ise %8 "eter!ine t*e ti!e-do!ain signal re.resented ' t*e (ollo;ing 5ourier series )oe((i)ients: ! k = j δ( k −2 )− j δ( k +2 )+δ( k − %)+δ( k + % ) 2B
(ω =# π ) $
Eer)ise 08 "eter!ine t*e ti!e-do!ain signal re.resented ' t*e (ollo;ing 5ourier series )oe((i)ients: ! k = j δ( k −2 )− j δ( k +2 )+δ( k − %)+δ( k + % )
(ω $ =0 π )
Eer)ise &8 "eter!ine t*e ti!e-do!ain signal re.resented ' t*e (ollo;ing 5ourier series )oe((i)ients: ∣k ∣
! k =(−2 / %)
(ω $ =2)
Eer)ise B8 "eter!ine t*e !agnitude and .*ase s.e)tra o( t*e (ollo;ing signal: n
x ( n)=(% / 0 ) u ( n− 0 ) 8 Eer)ise 68 "eter!ine t*e !agnitude and .*ase s.e)tra o( t*e (ollo;ing signal: ∣n∣
(∣a∣< 2 ) 8
x ( n)= a
Eer)ise 8 "eter!ine t*e !agnitude and .*ase s.e)tra o( t*e (ollo;ing signal: x ( n)=
2 / # + 2 / #)os ( π n / ' ) $
(∣n∣≤ ' ) (∣n∣> ' )
Eer)ise 38 "eter!ine t*e !agnitude and .*ase s.e)tra o( t*e (ollo;ing signal: x ( n)= # δ ( 0− # n ) 8 Eer)ise 3$8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing dis)reteti!e 5ourier trans(or!: ! (Ω)= )os ( # Ω)+ j sin ( # Ω) Eer)ise 328 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing dis)reteti!e 5ourier trans(or!:
(Ω / # ) ! (Ω)=sin (Ω)+ )os (Ω/ Eer)ise 3#8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing dis)reteti!e 5ourier trans(or!:
( π / 0 <∣Ω∣<% π / 0 ) (∣Ω∣≤π ∣≤π / 0 ∨∣Ω∣≥% π / 0 ) $ (∣Ω ar( { ! (Ω)}=−0 Ω
∣ ! (Ω)∣= 2
Eer)ise 3%8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: − # t
x ( t )= )= e
u ( t −%)
Eer)ise 308 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: − 0∣t ∣
x ( t )= e Eer)ise 3&8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: −t
x ( t )= te u ( t ) Eer)ise 3B8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: x ( t )=
∞
a ∑ =
k
δ ( t − k )
(∣a∣< 2 )
k $
Eer)ise 368 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!:
{
! (ω)=
)os ( # ω) $
(∣ω∣< π/ 0 ) (∣ω∣≥π/ 0 )
Eer)ise 38 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)=e
−# ω
(ω ) u (ω)
Eer)ise 338 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: 26
! (ω)=e
−#∣ω∣
Eer)ise 2$$8 "eter!ine t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal: −t
x ( t )= e )os ( # π t ) u ( t ) Eer)ise 2$28 "eter!ine t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal:
{
x ( n)=
)os ( π n / 2$ )+ j sin ( π n / 2$ ) $
(∣n∣<2$ ) (∣n∣≥2$ )
Eer)ise 2$#8 "eter!ine t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal: 2 +t
x ( t )= e u ( # −t ) Eer)ise 2$%8 "eter!ine t*e a..ro.riate 5ourier re.resentation o( t*e (ollo;ing signal: )=∣sin ( # π t )∣ x ( t )=∣ Eer) Eer)ise ise 2$08 2$08 "eter "eter!i !ine ne t*e t*e ti!e ti!e-do -do!a !ain in signa signall )orre )orres. s.on ondi ding ng to t*e t*e (oll (ollo; o;in ing g (re1uen)-do!ain re.resentation: − jk π/ #
! k =
(∣k ∣< 2$ ) (∣k ∣≥2$)
e
$
and t*e (unda!ental .eriod o( t*e signal T =2 8 Eer) Eer)ise ise 2$&8 2$&8 "eter "eter!i !ine ne t*e t*e ti!e ti!e-do -do!a !ain in signa signall )orre )orres. s.on ondi ding ng to t*e t*e (oll (ollo; o;in ing g (re1uen)-do!ain re.resentation:
{
! (ω)=
)os ( ω / 0 )+ j sin (ω / 0 ) $
(∣ω∣< π) (∣ω∣≥π)
Eer) Eer)ise ise 2$B8 2$B8 "eter "eter!i !ine ne t*e t*e ti!e ti!e-do -do!a !ain in signa signall )orre )orres. s.on ondi ding ng to t*e t*e (oll (ollo; o;in ing g (re1uen)-do!ain re.resentation: ! (Ω)=∣sin (Ω)∣ Eer)ise 2$68 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: −t
x ( t )= sin ( # π t ) e u ( t ) Eer)ise 2$8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: −%∣t − 2∣
x ( t )= te
Eer)ise 2$38 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: #sin ( % π t ) sin ( # π t ) x ( t )= π t π t Eer)ise 22$8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: d −# t x ( t )= te sin ( t ) u ( t ) dt Eer)ise 2228 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: t
sin ( # π τ ) τ ∫ π τ d τ −∞
x ( t )=
Eer)ise 22#8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: −t + #
x ( t )= e
u ( t − #)
Eer)ise 22%8 "eter!ine t*e 5ourier trans(or! o( t*e (ollo;ing signal: sin ( t ) d sin ( # t ) x ( t )= π t dt π t Eer)ise 2208 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)= 2
jω
( 2 + j ω )#
Eer)ise 22&8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)=
0sin ( # ω− 0 ) 0sin (# ω + 0 ) − # ω− 0 # ω +0
Eer)ise 22B8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)=
2 −πδ(ω) j ω ( j ω + # )
Eer)ise 2268 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)=
[
d sin ( # ω ) 0sin ( 0 ω ) ω d ω
]
Eer)ise 2#$8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)=
(ω ) #sin (ω) ω ( j ω + # )
Eer)ise 2#28 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 5ourier trans(or!: ! (ω)=
0sin
#
ω
(ω) (ω )
#
Eer)ise 2##8 >i4en t*e (ollo;ing 5ourier trans(or! .air:
{
x ( t )=
2 $
(∣t ∣< 2 ) (∣t ∣≥2 )
↔
! (ω)=
(ω ) #sin (ω) ω
E4aluate t*e 5ourier trans(or! o( t*e (ollo;ing signal:
)= x y ( t )=
t −# #
Eer)ise 2#%8 >i4en t*e (ollo;ing 5ourier trans(or! .air:
{
x ( t )=
2 $
(∣t ∣< 2 ) (∣t ∣≥2 )
↔
! (ω)=
(ω ) #sin (ω) ω
E4aluate t*e 5ourier trans(or! o( t*e (ollo;ing signal: y ( t )=sin (π t ) x ( t ) Eer)ise 2#08 >i4en t*e (ollo;ing 5ourier trans(or! .air:
{
x ( t )=
2 $
(∣t ∣< 2 ) (∣t ∣≥2 )
↔
! (ω)=
(ω ) #sin (ω) ω
E4aluate t*e 5ourier trans(or! o( t*e (ollo;ing signal: y ( t )= x ( t + 2 )− x ( # t −2) Eer)ise 2#&8 >i4en t*e (ollo;ing 5ourier trans(or! .air:
{
x ( t )=
2 $
(∣t ∣< 2 ) (∣t ∣≥2 )
↔
! (ω)=
(ω ) #sin (ω) ω
E4aluate t*e 5ourier trans(or! o( t*e (ollo;ing signal: y ( t )= # tx ( t ) Eer)ise 2#B8 >i4en t*e (ollo;ing 5ourier trans(or! .air: x ( t )=
{
2 $
(∣t ∣< 2 ) (∣t ∣≥2 ) 23
↔
! (ω)=
(ω)) #sin (ω ω
E4aluate t*e 5ourier trans(or! o( t*e (ollo;ing signal: y ( t )= x ( t ) x (t ) Eer)ise 2#68 >i4en t*e dis)rete-ti!e 5ourier trans(or! ! (Ω (Ω)) o( t*e (ollo;ing signal: ∣n∣
x ( n)= n ( % / 0 )
it*out e4aluating ! (Ω (Ω)) , (ind t*e signal y ( n ) i( its dis)rete-ti!e 5ourier trans(or!
(Ω)) is gi4en ': % (Ω − j 0 Ω
% (Ω)= e
(Ω)) ! (Ω
Eer)ise 2#8 >i4en t*e dis)rete-ti!e 5ourier trans(or! ! (Ω (Ω)) o( t*e (ollo;ing signal: ∣n∣
x ( n)= n ( % / 0 )
it*out e4aluating ! (Ω (Ω)) , (ind t*e signal y ( n ) i( its dis)rete-ti!e 5ourier trans(or!
(Ω)) is gi4en ': % (Ω (Ω)] % (Ω)=Re [ ! (Ω)] Eer)ise 2#38 >i4en t*e dis)rete-ti!e 5ourier trans(or! ! (Ω (Ω)) o( t*e (ollo;ing signal: ∣n∣
x ( n)= n ( % / 0 )
it*out e4aluating ! (Ω (Ω)) , (ind t*e signal y ( n ) i( its dis)rete-ti!e 5ourier trans(or!
(Ω)) is gi4en ': % (Ω % (Ω)=
(Ω)) d! (Ω d Ω
Eer)ise 2%$8 >i4en t*e dis)rete-ti!e 5ourier trans(or! ! (Ω (Ω)) o( t*e (ollo;ing signal: ∣n∣
x ( n)= n ( % / 0 )
it*out e4aluating ! (Ω (Ω)) , (ind t*e signal y ( n ) i( its dis)rete-ti!e 5ourier trans(or!
(Ω)) is gi4en ': % (Ω % (Ω)= ! (Ω)+ ! (−Ω) Eer)ise 2%28 >i4en t*e dis)rete-ti!e 5ourier trans(or! ! (Ω (Ω)) o( t*e (ollo;ing signal: ∣n∣
x ( n)= n ( % / 0 )
it*out e4aluating ! (Ω (Ω)) , (ind t*e signal y ( n ) i( its dis)rete-ti!e 5ourier trans(or!
(Ω)) is gi4en ': % (Ω % (Ω)=
d! ( # Ω) d Ω
Eer)ise 2%#8 A .eriodi) signal x ( t ) *as t*e (ollo;ing 5ourier series re.resentation: −∣k ∣
! k =−k k # #
it*out it*out deter!ining deter!ining x ( t ) , (ind (ind t*e 5ourie 5ourierr series series re.rese re.resenta ntatio tion n o( t*e (ollo; (ollo;ing ing signal:
y ( t )= x ( % t ) Eer)ise 2%%8 A .eriodi) signal x ( t ) *as t*e (ollo;ing 5ourier series re.resentation: −∣k ∣
! k =−k k # #
it*out it*out deter!ining deter!ining x ( t ) , (ind (ind t*e 5ourie 5ourierr series series re.rese re.resenta ntatio tion n o( t*e (ollo; (ollo;ing ing signal:
)= x ( t −2 ) y ( t )= Eer)ise 2%08 A .eriodi) signal x ( t ) *as t*e (ollo;ing 5ourier series re.resentation: −∣k ∣
! k =−k k # #
#$
it*out it*out deter!ining deter!ining x ( t ) , (ind (ind t*e 5ourie 5ourierr series series re.rese re.resenta ntatio tion n o( t*e (ollo; (ollo;ing ing signal: dx ( t ) dt *as t*e (ollo;ing 5ourier series re.resentation:
y ( t )= Eer)ise 2%&8 A .eriodi) signal x ( t )
−∣k ∣
! k =−k k # #
it*out it*out deter!ining deter!ining x ( t ) , (ind (ind t*e 5ourie 5ourierr series series re.rese re.resenta ntatio tion n o( t*e (ollo; (ollo;ing ing signal: y ( t )= )os ( 0 π t ) x ( t ) Eer)ise 2%B8 Set)* t*e !agnitude res.onse and t*e .*ase res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse: − # t
h ( t )=δ( t )− # e
u ( t )
Eer)ise 2%68 Set)* t*e !agnitude res.onse and t*e .*ase res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse: − # t
h ( t )= 0 e
)os ( &$ t )
Eer)ise 2%8 Set)* t*e !agnitude res.onse and t*e .*ase res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse:
()
2 6 h ( n )= C C
n
u(n)
Eer)ise 2%38 Set)* t*e !agnitude res.onse and t*e .*ase res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing i!.ulse res.onse:
{
(−2 )n (∣n∣≤2$ ) h ( n )= (∣n∣> 2$ ) $ Eer)ise 20$8 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( a sste!, gi4en t*e (ollo;ing .air o( in.ut and out.ut signals (or t*e sste!: − t
− # t
x ( t )= e u ( t ) and y ( t )= )= e
− % t
u ( t )+ e
u ( t )
Eer)ise 2028 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( a sste!, gi4en t*e (ollo;ing .air o( in.ut and out.ut signals (or t*e sste!: − % t
x ( t )= e
−%( t −# )
u ( t ) and y ( t )= e
u ( t −# )
Eer)ise 20#8 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( a sste!, gi4en t*e (ollo;ing .air o( in.ut and out.ut signals (or t*e sste!: − # t
x ( t )= e
−# t
u ( t ) and y ( t )= # te
u ( t )
Eer)ise 20%8 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( a sste!, gi4en t*e (ollo;ing .air o( in.ut and out.ut signals (or t*e sste!: n
n
n
x ( n)=( 2 / #) u ( n ) and y ( n )=(2 / 0)( 2 / # ) u ( n)+ )+((2 / 0 ) u ( n )
Eer)ise 2008 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( a sste!, gi4en t*e (ollo;ing .air o( in.ut and out.ut signals (or t*e sste!: n
n
x ( n)=( 2 / 0 ) u ( n) and y ( n )=(2 / 0) u ( n )−( 2 / 0)
n− 2
u ( n − 2)
Eer)ise 20&8 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: dy (t ) + % y ( t )= x ( t ) dt
#2
Eer)ise 20B8 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
−dx ( t ) dy ( t ) + B y ( t )= dt dt
+&
Eer)ise 2068 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
)−(( 2 / ) y ( n −# )=% x ( n )− )−(( % / 0 ) x ( n −2 ) y ( n )−(2 / 0) y ( n−2 )− Eer)ise 208 "eter!ine t*e (re1uen) res.onse and t*e i!.ulse res.onse o( t*e sste! des)ri'ed ' t*e (ollo;ing di((eren)e e1uation:
)+(( 2 / # ) y ( n−2 )= x ( n )− # x ( n −2) y ( n )+ Eer) Eer)ise ise 2038 2038 "eter "eter!i !ine ne t*e t*e ti!e ti!e-do -do!a !ain in signa signall )orre )orres. s.on ondi ding ng to t*e t*e (oll (ollo; o;in ing g unilateral La.la)e trans(or!: 2
2
! ( s )=
( s + # )( s + % )
Eer) Eer)ise ise 2&$8 2&$8 "eter "eter!i !ine ne t*e t*e ti!e ti!e-do -do!a !ain in signa signall )orre )orres. s.on ondi ding ng to t*e t*e (oll (ollo; o;in ing g unilateral La.la)e trans(or!: − # s
2
! ( s )=e
[
d 2 ds ( s + 2)#
]
Eer) Eer)ise ise 2&28 2&28 "eter "eter!i !ine ne t*e t*e ti!e ti!e-do -do!a !ain in signa signall )orre )orres. s.on ondi ding ng to t*e t*e (oll (ollo; o;in ing g unilateral La.la)e trans(or!: 2
2
! ( s )=
( #s +2) #+ 0 Eer)ise 2 >i4en t*e La.la)e trans(or! ! ( s ) o( t*e (ollo;ing signal: x ( t )= )os ( # t ) u ( t ) "eter!ine t*e signal y ( t ) i( its La.la)e trans(or! is gi4en ': )=(( s + 2 ) ! ( s ) % ( s )= Eer)ise 2&%8 >i4en t*e La.la)e trans(or! ! ( s ) o( t*e (ollo;ing signal: x ( t )= )os ( # t ) u ( t ) "eter!ine t*e signal y ( t ) i( its La.la)e trans(or! is gi4en ': ( % s ) % ( s )= ! ( Eer)ise 2&08 >i4en t*e La.la)e trans(or! ! ( s ) o( t*e (ollo;ing signal: x ( t )= )os ( # t ) u ( t ) "eter!ine t*e signal y ( t ) i( its La.la)e trans(or! is gi4en ': − % ( s )= s ! ( ( s ) Eer)ise 2&&8 >i4en t*e La.la)e trans(or! o( t*e signal x ( t ) as (ollo;s: #
# s ! ( s )= # s + # "eter!ine La.la)e trans(or! o( t*e (ollo;ing signal: y ( t )= x ( % t ) Eer)ise 2&B8 >i4en t*e La.la)e trans(or! o( t*e signal x ( t ) as (ollo;s: # s ! ( s )= # s + # "eter!ine La.la)e trans(or! o( t*e (ollo;ing signal:
y ( t )= x ( t −# ) ##
Eer)ise 2&68 >i4en t*e La.la)e trans(or! o( t*e signal x ( t ) as (ollo;s: # s ! ( s )= # s + # "eter!ine La.la)e trans(or! o( t*e (ollo;ing signal: y ( t )= x ( t )
dx ( t ) dt
Eer)ise 2&8 >i4en t*e La.la)e trans(or! o( t*e signal x ( t ) as (ollo;s: # s ! ( s )= # s + # "eter!ine La.la)e trans(or! o( t*e (ollo;ing signal: −t
y ( t )= e x ( t ) Eer)ise 2&38 >i4en t*e La.la)e trans(or! o( t*e signal x ( t ) as (ollo;s: # s ! ( s )= # s + # "eter!ine La.la)e trans(or! o( t*e (ollo;ing signal:
)= # tx ( t ) y ( t )= Eer)ise 2B$8 >i4en t*e (ollo;ing La.la)e trans(or! .air: −at
x ( t )= e
u ( t )
↔
2 ! ( s )= s + a
E4aluate t*e unilateral La.la)e trans(or! o( t*e (ollo;ing signal: − at
)= e y ( t )=
)os ( ω$ t ) u ( t )
Eer)ise 2B$8 "eter!ine t*e (or)ed and natural res.onses (or t*e LTI sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation ;it* t*e s.e)i(ied initial and in.ut )onditions: )onditions: dy (t ) + 2$ 2$ y y ( t )=2$ 2$ x x ( t ) , y ( $ − )=2 , and x ( t )= u ( t ) dt Eer)ise 2B28 "eter!ine t*e (or)ed and natural res.onses (or t*e LTI sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation ;it* t*e s.e)i(ied initial and in.ut )onditions: )onditions: #
d y ( t ) #
dt
+&
∣
dy ( t ) dx ( t ) dy ( t ) , y ( $ −)=−2 , + B y ( t )= )=− −0 x ( t )−% dt dt dt
=& and
t = $−
−t
x ( t )= e u ( t ) Eer)ise 2B#8 "eter!ine t*e (or)ed and natural res.onses (or t*e LTI sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation ;it* t*e s.e)i(ied initial and in.ut )onditions: )onditions: #
d y ( t ) #
dt
+ y (t )= C x (t ) , y ( $ − )=$ ,
∣
dy ( t ) dt
=# and x ( t )= e−t u ( t )
t = $−
Eer)ise 2B%8 "eter!ine t*e (or)ed and natural res.onses (or t*e LTI sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation ;it* t*e s.e)i(ied initial and in.ut )onditions: )onditions: #
d y ( t ) #
dt
+#
∣
dy ( t ) dx ( t ) dy ( t ) , y ( $ − )= # , + & y (t )= dt dt dt
=$ and x (t )= u (t )
t = $−
Eer)ise 2B08 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 'ilateral La.la)e trans(or!: & s
e ( Re ( s )<−# ) ! ( s )= s + # Eer)ise 2B&8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 'ilateral La.la)e trans(or!: #%
#
! ( s )=
d
( ) 2
(Re ( s )> % ) # ds s −% Eer)ise 2BB8 "eter!ine t*e ti!e-do!ain signal )orres.onding to t*e (ollo;ing 'ilateral La.la)e trans(or!:
− s −0 ! ( s )= # (−# < Re ( s )<− 2) s + % s + # Eer)ise 2B68 "eter!ine t*e i!.ulse res.onse o( a )ausal sste! *a4ing t*e (ollo;ing trans(er (un)tion: " ( s )=
#
+ # s − # # s − 2
# s
Eer)ise 2B8 "eter!ine t*e i!.ulse res.onse o( a sta'le sste! *a4ing t*e (ollo;ing trans(er (un)tion: " ( s )=
#
+ # s − # # s − 2
# s
Eer)ise 2B38 "eter!ine t*e i!.ulse res.onse o( a )ausal sste! *a4ing t*e (ollo;ing trans(er (un)tion: # s −2 " ( s )= # s + # s + 2 Eer)ise 26$8 "eter!ine t*e i!.ulse res.onse o( a sta'le sste! *a4ing t*e (ollo;ing trans(er (un)tion: # s −2 " ( s )= # s + # s + 2 Eer)ise 2628 "eter!ine t*e i!.ulse res.onse o( a )ausal sste! *a4ing t*e (ollo;ing trans(er (un)tion: −& s
" ( s )= e
+
#
s − #
Eer)ise 26#8 "eter!ine t*e i!.ulse res.onse o( a sta'le sste! *a4ing t*e (ollo;ing trans(er (un)tion: −& s
" ( s )= e
+
#
s − #
Eer)i Eer)ise se 26%8 26%8 "eter! "eter!ine ine t*e trans(er trans(er (un)ti (un)tion on and t*e i!.uls i!.ulsee res.ons res.onsee o( a sta'le sta'le sste!, gi4en a .air o( its in.ut and out.ut signals as (ollo;s: − t
− # t
x ( t )= e u ( t ) and y ( t )= e
)os ( t ) u ( t )
Eer)i Eer)ise se 2608 2608 "eter! "eter!ine ine t*e trans(er trans(er (un)ti (un)tion on and t*e i!.uls i!.ulsee res.ons res.onsee o( a sta'le sta'le sste!, gi4en a .air o( its in.ut and out.ut signals as (ollo;s: x ( t )= e
− # t
−t
− % t
u ( t ) and y ( t )=−# e u ( t )+ # e
u ( t )
Eer)ise 26&8 "eter!ine t*e trans(er (un)tion and t*e i!.ulse res.onse o( a )ausal sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: dy (t ) + 2$ 2$ y y ( t )=2$ 2$ x x ( t ) dt Eer)ise 26B8 "eter!ine t*e trans(er (un)tion and t*e i!.ulse res.onse o( a )ausal sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
+&
dy ( t ) dx ( t ) + B y ( t )= x ( t )+ dt dt #0
Eer)ise 2668 "eter!ine t*e trans(er (un)tion and t*e i!.ulse res.onse o( a )ausal sste! des)ri'ed ' t*e (ollo;ing di((erential e1uation: #
d y ( t ) #
dt
−
dy ( t ) dx ( t ) −# y (t )=−0 x (t )+ & dt dt
Appro2ed 34
Appro2ed 34
Appro2ed 34
Un2er!)4 Board
Fac'.)4 Board
Depar)men)
#&
Prepared 34
