Ss Answers Chapter1

8/11/2019 Ss Answers Chapter1

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Signals & Systems(Second Edition)

Learning Instructions

(Exercises Answers)

Department of Computer Engineering2005.12

0


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ContentsChapter 1 2

Chapter 2 17

Chapter 3 53

Chapter 4 80

Chapter 5 101

Chapter 6 127

Chapter 7 137

Chapter 8 150

Chapter 9 158

Chapter 10 178

Chapter 1 Answers1.1 Converting from polar to Cartesian coordinates:

1 1 1os

2 2 2

j

e

= = 1 1 1

os! "2 2 2

j

e

= =

1


3/17

2 os! " sin! "2 2

jj je

= + = 2 os! " sin! "

2 2

jj je

= =

5

2 2j j

je e

= = 4 2 !os! " sin! " " 14 42

j

j je

= + = +

9

4 4 12 2j j

je e

= = 9

4 4 1

2 2

j j

j

e e

= =

4 12j

je

=

1.2 converting from Cartesian to polar coordinates:0

5 5 j

e= # 2 2 j

e = # 23 3

jj e

=

21 3

2 2

jj e

= # 41 2 jj e

+ = # ( )2

221 j

j e

=

4!1 "j j e

= # 41

1

j

j e

+=

12

2 2

1 3

j

j e

+ =+

1.3. !a" E $ 4

0

1

4

tdte

= # P $0# %ease E <

!%" !2 "42! "

j ttx e

+= # 2! " 1tx = .'here(ore# E $

2

2! " dttx+

$ dt+

$ #P $

21 1lim lim22 2

! "T T

T TT Tdt dt

T Ttx = = lim1 1T =

!"2! "tx $os!t". 'here(ore# E $

2

3! " dttx+

$2

os! " dtt+

$ #

P $21 1 1 !2 " 1

lim lim2 2 2 2

os! "T T

T TT T

COS t dt dt

T Tt

+= =

!)" 1* + * +1

2

n

n u nx =

#2

* +1

1* +

4

n

u nnx = . 'here(ore# E $

2

0

41 3

1* +

4

n

n

nx+

=

= =

P $0%ease E , .

!e" 2* +nx $ ! "

2 8nj

e + # 22* +nx $1. there(ore# E $

2

2* +nx+

$ #

P $21 1

lim lim 1 122 1 2 1* +

N N

N Nn N n N N N

nx = =

= =+ + .

!(" 3* +nx $os4

n

. 'here(ore# E $2

3* +nx+

$

2

os! "4

n+

$

2

os! "4

n+

#

P $ 1

lim os2 1 4

n

N

Nn NN

=

=+

1 os! "1 12lim ! "

2 1 2 2

N

Nn N

n

N

=

+=

+ 1.4. !a" 'he signal -*n+ is shi(te) %y 3 to the right. 'he shi(te) signal will %e ero (or n,1# An) n/7.

!%" 'he signal -*n+ is shi(te) %y 4 to the le(t. 'he shi(te) signal will %e ero (or n,6. An) n/0.!" 'he signal -*n+ is (lippe) signal will %e ero (or n,1 an) n/2.

!)" 'he signal -*n+ is (lippe) an) the (lippe) signal is shi(te) %y 2 to the right. 'he new Signal will %e ero(or n,2 an) n/4. !e" 'he signal -*n+ is (lippe) an) the (lippe) an) the (lippe) signal is shi(te) %y 2 to the le(t.

'his new signal will %e ero (or n,6 an) n/0.1.5. !a" -!1t" is o%taine) %y (lipping -!t" an) shi(ting the (lippe) signal %y 1 to the right.

'here(ore# - !1t" will %e ero (or t/2.!%" rom !a"# we now that -!1t" is ero (or t/2. Similarly# -!2t" is ero (or t/1#

'here(ore# - !1t" -!2t" will %e ero (or t/2.!" -!3t" is o%taine) %y linearly ompression -!t" %y a (ator o( 3. 'here(ore# -!3t" will %e

ero (or t,1.!)" -!t3" is o%taine) %y linearly ompression -!t" %y a (ator o( 3. 'here(ore# -!3t" will %e

ero (or t,9.1.6 !a"x1!t" is not perio)i %ease it is ero (or t,0.

!%"x2*n+$1 (or all n. 'here(ore# it is perio)i with a (n)amental perio) o( 1.!"x3*n+ is as shown in the igre S1.6.

2

3

1

4

1

1

0

4

111

1

n

5

x3*n

+


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'here(ore# it is perio)i with a (n)amental perio) o( 4.1.7. !a"

( )1

* +v

nx $ { }1 11 1

* + * + ! * + * 4+ * + * 4+"2 2

n n u n u n u n u nx x+ = +

'here(ore# ( )1

* +v

nx is ero (or 1* +nx /3.

!%" Sine x1!t" is an o)) signal# ( )2

* +v

nx is ero (or all ales o( t.

!" ( ) { }1 13

1 1* + * + * + * 3+ * 3+

2 2

1 1

2 2v

n n

n n n u n u nx x x

= + =

'here(ore# ( )3* +v

nx is ero when n ,3 an) when n .

!)" ( )1

5 5

4 4

1 1! " ! ! " ! "" ! 2" ! 2"

2 2v

t tt t t u t u t x x x e e

= + = +

'here(ore# ( )4! "

v

tx is ero only when t .

1.8. !a" ( ) 0

17 ! "8 2 2 os!0 "

tt tx e = = +l

!%" ( ) 0

2 ! " 2 os! " os!3 2 " os!3 " os!3 0"

4

tt t t t x e

= + = = +l

!" ( )3 ! " sin!3 " sin!3 "2t t

t t tx e e

= + = +l

!)" ( ) 2 2

4 ! " sin!100 " sin!100 " os!100 "

2

t t tt t t t x e e e

= = + = +l

1.9. !a"1

! "tx is a perio)i omple- e-ponential.

10 10 21! " j t j t t jx e e

+ = =

!%" 2! "tx is a omple- e-ponential mltiplie) %y a )eaying e-ponential. 'here(ore#2! "tx is not perio)i.

3* +nx is a perio)i signal. 3* +nx $ 7j n

e

$ j n

e

.

3* +nx is a omple- e-ponential with a (n)amental perio) o(2

2

= .

!)" 4* +nx is a perio)i signal. 'he (n)amental perio) is gien %y $m!2

3 5

"

$ 10! ".

3m :y hoosing m$3. ;e o%tain the (n)amental perio) to %e 10.

!e" 5* +nx is not perio)i. 5* +nx is a omple- e-ponential with 0w $35. ;e annot (in) any integer m sh

that m!0

2

w

" is also an integer. 'here(ore# 5* +nx is not perio)i.

1.10. x!t"$2os!10t1"sin!4t1"

S $2

10 5

= .

S $2

4 2

= .

'here(ore# the oerall signal is perio)i with a perio) whih the least ommonmltiple o( the perio)s o( the (irst an) seon) terms. 'his is e?al to .

1.11. -*n+ $ 1+ 7

4j n

e @

2

5j n

e

S $1.

S $

74

2$7 when m$2

3


5/17

0123 1 2 3

*n+

n

igre S 1.12

1

0

1

2101 t

1

2

g!t"

2

3 3

t

igre S 1.14

x!t"

S $

52

2

$5 !when m$1"

'here(ore# the oerall signal -*n+ is perio)i with a perio) whih is the least ommonBltiple o( the perio)s o( the three terms inn -*n+.'his is e?al to 35.

1.12. 'he signal -*n+ is as shown in (igre S1.12. -*n+ an %e o%taine) %y (lipping *n+ an) then

Shi(ting the (lippe) signal %y 3 to the right. 'here(ore# -*n+$*n3+. 'his implies that B$1 an) no$3.

1.13

y!t"= t

dtx "! $ dtt

""2!"2!! + $

>

ere# ! " ! ".xy xxt t T = an) ! " ! ".yy xxt t =

1.38.!a" ;e now that 22 !2 " ! ".t t =V V 'here(ore'his implies that

1!2 " ! ".

2t t =

!%"'he plot are as shown in igre s3.18.1.39 ;e hae

0 0lim ! " ! " lim !0" ! " 0.u t t u t

= =

V VV V

Also#

13

0 02 2g)! # " g)! # " g)! # " g)! # ".T

! ! ! !! !

= = =

2

1lim !2 " lim ! ".

2t t

=V V

V V

0

1lim ! " ! " ! ".

2u t t t =V VV

1

112et

O

P t

1

OO0 t

O

P t

1

12

OO

0

12et

t

O

P t

1

12

O2O2 t0t

O

P t

1

2

O

Ot0t

O

P t

112

OO t0

t

O

P t

112

OO

t0

t


15/17

;e hae

==

0"!"!"!"!"! dtudtutg

'here(ore#

0# 0

! " 1# 0

0

t

g t t

unde"ined "or t

>=


16/17

( ) "! TtyTtx s ow i( - !t" is perio)i with perio) '. -t$-!t'". 'here(ore# we may onl)e that y!t"$y!t'".'his impliesthat y!t" is also perio)i with ' .A similar argment may %e ma)e in )isrete time .!%"1.44!a" Assmption F ( -!t"$0 (or t,t0#then y!t"$0 (or t, t0.'o proe 'hat F 'he system is asal.

Eet s onsi)er an ar%itrary signal -1!t" .Eet s onsi)er another signal -2!t" whih is the same as -1!t"(or t, t0.:t (or t/ t0# -2!t"-1!t"#Sine the system is linear#

( ) ( ) ( ) ( )1 2 1 2 #x t x t y t y t

Sine ( ) ( )1 2 0x t x t = (or t, t0 #%y or assmption $ ( ) ( )1 2 0y t y t = (or t, t0 .'his implies that

( ) ( )1 2y t y t= (or t, t0. n other wor)s# t he otpt is not a((ete) %y inpt ales (or 0t t . 'here(ore# thesystem is asal .AssmptionF the system is asal . 'o proe that F( -!t"$0 (or t, t0.then y!t"$0 (or t, t0.

Eet s assme that the signal -!t"$0 (or t, t0.'hen we may e-press -!t" as ( ) ( )1 2! "x t x t x t= #

;here ( ) ( )1 2x t x t= (or t, t0. the system is linear .the otpt to -!t" will %e ( ) ( )1 2! "y t y t y t= .ow

#sine the system is asal . ( ) ( )1 2y t y t= (or t, t0.implies that

( ) ( )1 2y t y t= (or t, t0.'here(ore y!t"$0 (or t, t0.!%" Consi)er y!t"$-!t"-!t1" .ow # -!t"$0 (or t, t0 implies that y!t"$0 (or t, t0 .ote that the system isnonlinear an) nonasal .!" Consi)er y!t"$-!t"1. the system is nonlinear an) asal .'his )oes not satis(y the on)ition o( part!a".!)" AssmptionF the system is inerti%le. 'o proe that Fy*n+$0 (or all n only i( -*n+$0 (or all n .Consi)er

* + 0 * +x n y n= .Sine the system is linear F

2 * + 0 2 * +x n y n= .Sine the inpt has not hange) in the two a%oe e?ations #we re?ire that y*n+$2y*n+.'his implies that y*n+$0. Sine we hae assme) that the system is inerti%le #only one inpt ol) hae le) to this partilar otpt .'hat inpt mst %e -*n+$0 .AssmptionF y*n+$0 (or all n i( -*n+$0 (or all n . 'o proe that F 'he system is inerti%le .Sppose that

1 1* + * +x n y n

an)

2 1* + * +x n y n

Sine the system is linear #

1 2 1 2* + * + * + * + 0x n x n y n y n = =

:y the original assmption #we mst onl)e that 1 2* + * +x n x n=

.'hat is #any partilar y1*n+ an %e pro)e)that %y only one )istint inpt -1*n+ .'here(ore # the system isinerti%le.!e" y*n+$-2*n+.1.45. !a" Consi)er #

( )11 1

! " ! "s

hxx t y t t =

an)

( )22 2

! " ! "s

hxx t y t t = .

ow# onsi)er ( ) ( ) ( )3 1 2x t ax t bx t= + . 'he orrespon)ing system otpt will %e

15


17/17

( ) ( )1 2

3 3

1 2

1 2

! " ! " ! "

! " ! " ! " ! "

! " ! "

hx hx

y t x h t d

a x h t d b x t h t d

a t b t

ay t by t

= +

= + + +

= +

= +

'here(ore# S is linear .ow #onsi)er -4!t"$-1!t'".'he orrespon)ing system otpt will %e

( )1

4 4

1

1

! " ! " ! "

! " ! "

! " ! "

hx

y t x h t d

x T h t d

x h t T d

t T

= +

= +

= + +

= +

Clearly# y4!t"y1!t'".'here(ore #the system is not timeinariant.'he system is )e(initely not asal %ease the otpt at any time )epen)s on (tre

ales o( the inpt signal -!t".!%" 'he system will then %e linear #time inariant an) nonasal.1.46. 'he plots are in igre S1.46.1.47.!a" 'he oerall response o( the system o( igre

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Ss Answers Chapter1