Signals systems and transforms 5th edition phillips solutions manual by bitech98

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t I

Chapter 2 solutions 2.1 (a)

(i)

.r(-t/3)

-9 -8 - 7 -G -5 -4 -3 -2 -1

5 6

7 8

(ii)

-4 -(p

'?:'=- 3t-t,.

-.3 VJ

2..

�

7 l+z.=t

,X.(3t - (p)

-1 0

t=="'� +2

:>t

�. \

7Ro�L61V\.

��

�

T D"Y\ 1;i 'Au -e A

• ...L, !-c

-�

.:r(3

( l,bJ.. )

i-r-•

t.)

5 -l

-I I

-D-8-7-G-5-4-3-2-1

-l

...

f t .

•

l f

W-,

� I

,. I

...•..

_..

1---l .. J

(iv) +

J.'(2 - t) 5

-1

•

.....

1 2 3 -! 5

;�

•

X(-6)

ti.") Jt (--f:/-=,

:ter ')

=;'>

--t-;3

-=)

-3 "t'

-So.

•

(1,c'.) ,x(�t-�) -=?Cli:-\ =l L

=3/:::-t,,

-z;i:.

?((3t-t, ')

(-2,(.:/')

X(3>+ t: ') ::.. ;t:, { ?:' 'J �) 'C"' = t + 3

1.:u-) X (J- t ') ==? T-=

;t-�

�t

= 'Z:""- 3

:;> t- = -1: +- 2X(2-�)

-1

-= 'r-a+2

.3 2

-'i

-�

-3

.. l

-27

c��)

')G{:it-h)

= 'X. £ r"\

-=)

c- = 3t-Ct,

�

. 7l(�t.-b ')

2 1

•

-3

().,. ;.�)

X(:i+t')

t= (-1-s, =-

c/3

+ 2..

.. -

-2. -(

= �(i-') =? T-:-3+t .:ry t z; ,Z--- 3

..---------'A-3 x:...

( 3 + t: )

2.--

-12

(iu-)

-q

-�

-3

"X- { 2- t: J -=- x.Jr:)

�

-r:: ;). - t:

\2.

-;> C z: -1: +- 2...

,%(2-t) •

.,2.

-�--+-� ._+---l-1-----+--1 -$ -1.R --4 -1. A,

(--4---'fl---f--+----....

�

1 I

VR.DB LE:IV\..

�I?,.

(0..\

1 I

I [

•

t I

+ I

l l

+ +

f I

...

....

�

l- -'

�.ttt.°')+2

-·

-3

I +

(1L')

-+ 1

(i ')

-3

+..

L �

j_ -; I- �-

j -3 -

2. 'X.(�·f)+ 2

(it?)

-4X.1.c')+2 2.

-'3

-1,S

t,5

-4 -i.

j II

J �

-2.

(�i)

._.

f l

4x.Lf::) -z g

G:,

'2..

x. (�-i)

-2

-I

-3 =t»

-z...

2 j 4-,1:

�-----lf-----•

-- 2.

� ::. ?J(.+ 2..

-z..

_.,

-2.

-z.

-2.

. �::: �X.+z.

�,

2,x,{-i_€)+2..

�2.

2..

-2.

11-

I()

. -t!f14--i-

-4,

-l

8' lp

'-f

-2...

2...

=-112

2..

�) t

e: �

'9 '('"" � �t-

� l t' :: 'A. "K.{2.t)+z.,

(iL�)

-2.

-1 -lp

P�.Q(C.') (.i.) j {'t')

4 �lt') - 2 "./

'1' z: t 8'

2..

t 0

7. -2...

-l

-0

�ti;{:)-2..

-..}

'.J

&, 4·

,2.

--fti

-3

-2

•

[.l�)tj(-r)=

';)."i,(-i:-)+-L

�

.2...

'0 't"" :;. t '4:-t.. 4 2.. �

-1

- G:,

(i-i.-l-) �(-r:--'-= J. �[2.t')-1- 2 �

�<LL-tj+Z..

-=)?-=')..t

-3

lLJ

7 t: = "t'/2-

;l.�+-2.

'B'

2. \

{p'

-t

"2.

-1

-3

{?V-} �('[):: - 4�(t )-tZ.

t, 5

-1,C,

-1()

-(c,

-2 2.

� 1:-:...t

·-41..lt\ + 2..

-·U'. -t-2

3 z,

;t :t(�c. -r-2

-(.p

-3

-z.. -1

-�

-7

-10

4,?

-I

-2l·(t)

-2:1:( -2t + 2)

3 .

2 l

-3

-1 -t

•

-2

(a) y(t)

-2(i·(-2t + 2))

(b)

t -0.5 -1

y(t) 4 2

0.-1

-2t+2 3 -1 0

-2(x(-2t - 1)) + 2 4 2 0.4

•

:So f u+�trvi

(t;t)

P f<.., 4 _

8 (f; ��-�-x.(- f:./2_ -y 3 2.

7: :: --t-/2

-z...

+2.� +-4-

+Z,

:.)

�=-22C.+4-

= - 2 t: + 4 v'

z· 4

-1

+z,

�

-J

•• +

• •.'..

:jt- i• I+

_J -

7-3 _3.

J_L_L__l -��L--

�5

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0 1- -J,5

+ -

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, -

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j.-+- Al

j � _._ .

j �

f 'rO b (e. W\. . t-

t (-t)

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· tf �e(t)

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x {-t)

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3+ 3... .3+

3 .3

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j . �

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t='k;0-6 l:e

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VV'l

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Ii I

�

1 1 . I �) ;ll-L'\ :: - +t: � ';l{-t) = 4-t .l - j 1· - - i X (-t ') = � C- -t ') .', "i { -t- ') i's o d d · . +l l f -1[b) XC-t) = e-H:\. � 'Jl(-t) =-e-/-t/ = e-/tl I . , , , ++

_j :

.. t �-

L · - -, -

� (t) == � Cos 8 i:

(_O') �

�

l .

� !

-l.

Ld) . X C.

-t -

�

-= 5,�-3t

')-=-

� lr) ,� _ e

�

X. C-t")

-:=. �

(-t ) .

'f(t)::= - �(ot) , x. (.+") ts e�

(e) X.L-t-)== ult)-u(--t) -=/t..(-t)=u(-t.'J-4(-t') .!1 -=- -[u.lt"> -u.l-t-)] t 1: (:t) X.G-t) : , 1...(t) ls ddd. j

...l -

·- .

=- 5&-o.-.3t

-= �( 3[t+�j)

��t + ��

I 1,

� X{-t ') � 5 Cds 3 (-t')

t: {&; =- � (- t) .: ,

1'.'. ( t ) = -:t C - t ) :. 'Jl {t ) l :; e treY1 · '. _[ •

l+"i

'X.(t\

-=-·- U.(t-1\ T IA.(-t-i)

")(. (-t) � - it (-t-1) � { - t)

= -'Xli-)

)

-f-

u.(t-1)

r,� O

�

(+) i 5

•

(e) x0(0)=-Xo(-O)= -x0(0). The only number with a=-a is azO so this implies Xo(O)=O. x(0)9ee(O)+x0(0)9ee(O) .

â&#x20AC;˘

(a) Let z(t) be the sum of two even functions x1(t) and x2(t). To show that .:(t) is even. we need to show that z (t) = z(-t) for all t. This ·is easy to show, since z( t) = xi ( t) + 1·2( t) and .:( -t) = l'i( -t) + :i.•2 (-t) (since to get z(-t) we just plug in -t everywhere fort, which amounts to just plugging in -t in 1·1(t) and 1·2(t)). Xow since 1:i(t) and 1·2(t) are even. by definition xi(t) = x1(-t) and :r2(t) = 1·2(-t) so x1(t) + :r2(t) = x1(-t) + x2(-t) so z(t) = z(-t). {b) Let x1(t) and 1·2(t) he two odd functions. Then 1·1(-t) x2(t)) which shows that 1'1 (t) + J:2(t) is odd.

+ x2(-t) =

-x1(t)

+ (-x2(t)) = -(l'1(t) +

(c) Let z(t) = x1(t) + 1)2(t) as in part a, where now x1(-t) = x1(t) and xz(-t) = -1·2(t). \Ve need to show that z(t) 'F z(-t): z(t) f -z(-t). Consider that z(-t) = xi (-t) + :1·2(-t) = 1·1 (t) - ;1.·2(t). In order to have .:(t) be even: we would therefore need to have 1·1 (t) + 1·2(t) = x1 (t) -1·2(t) for all t . which is equivalent to having 1:2(t) = -1:2(t) for all t, which is not possible for nonzero l'.2(t). Similarly, in order to have :(t) be odd. we would need to have z(t) = -z(t) � xi (t) + 1·2(t) = 1·2(t) - xi (t). which is not possible for nonzero x1(t). So the sum of an even and odd function must be neither even nor odd. (d) Let z(t) = x1(t)x2(t) where :t1(t.) = J:1(-t) and x2(t) x1(t)x2(t) = z(t) which shows that -�(t) is even. (e) Let .:(t) = 1:i(t)x2(t). where X[(t) .:(-t) = :ri(-t)x2(-t) = (-1·i(t)) (-x2(t))

= 1·1(-t)x:?(-t) =

Then z(-t) = 1·1(-t.)1·2(-t) =

-xi(-t) and x2(t} = -x2(-t). Clearly .:(t) is even because 1·i(t)1·2(t) = .:(t)� which is the definition of evenness.

(t) = -1·i(-t) and x2(t) = 1·2(-t). Clearly :;(t) is odd because (-x1 (t)) 1·2(t) = -xJ(t).r2(t) = -:(t). which is the definition of oddness.

(f) Let :;(t) = l'1(t):r2(t), where .:(-t)

= 1·2(-t).

-x.L+\ i 2.-

�c)

C-t)

=- "l{-t-> - r..C-t"> 2..

x.e : : %l-t \) + 7C (-t) 2:...

••

•

�

T -i j I

I-

-,.

1-L I

j- -

PR.f'-.R> l 6 fl\_

� .. \

l . J

+--!-

- 1

�

l-, ---,...

--'--+

(a) sin(t) = sin(t + n2rr) for any integer n, so 7sin(3t) = 7sin(3t + n2;r) = 7sin {3(t +n2:r)); therefore l'(t) is periodic with fundamental period To=� and fundamental frequency wo = = 3.

7-)

(b) sin(8(t + + �O) = sin(8t + 21r + 30) = sin(8t + 30). wo = 8 and To = 7 = ! . (c) eJt = cos(t) + j sin(t) is periodic with fundamental period 2rr: so e121 is periodic with fundamental period � = 1r. and fundamental frequency wo = 2. J

I �-i:+ �Gt JV _ � T -=. TI - 2 - II J 2-

(c\) �

Tc;={(l'T\: .. To

c_e')

�'tr -:::.... Ti 5 -t 12,

= .1]: -:= 5J /LCt_:ft o- o-/ ol% '2.. 111fe�

'61'�')) -r�L.) I

� =- z

-J ( D t- -t--11!�' -iv� -d rc, t . . , -J r•'l L � = e e -=- �-rr;� -1��1 e =-(o. '5 + j D�8{.p{p) e-},ot . f

T0;:

�:

�

= 1f7'5( �; �&-G,

•

'P �, i I c�- -x.. t-t-)

:: �

Cb) ')( C-C)-=

-- C� '\ ld1 (e')

i(..

ot -1- Al-wl5t

�t:;-f � ri-'-6

C--+:) =- � 3-t

+ -� �-6

��n-'-t +

>G(-+') =

n-t) : :

� 47f-l:

+ CKL(St)

�41Ft -!-�$Ji,-t + <2,a,�1Tt

�C'-t)-=: �(Si:-

c+ 1

+3D0)

e a-2-t: +�L3rrr-&')

:hl)-r-.

� { IA••

(a..)

-r,

-=-

ID ::

•

;nr �

:> Tl

T2-

i?.1.1

;nr 5

)

s Tu - -� � ,- , -y·

/� d< 0

�=----------------

...

•

l.b)

-l, {+')-=: � � $.:,/!: � 1

P�t- �

Y.2-- Lf:) := .5e�'!:, ·t:;") == � y "'(:" Tj -=.. ;;;?_ Ty T2 ::: .). 'Tt/ r-

//'"{

;;r-Jt-2-·�

�d�

-:I'S

�ci:'c..-

)

T-�. :::.

t?.5�;() T�3 �

cc J

" _.

7/�--

.e; diwc<J-Tk-=5

--l -i... :zta

Ti -

b...

�

i-r17�

f..12 �<::: � 'lT'

r;:- ccn. 4rr-t (d,) -n��er {=t::z) ..;(;O

.Lj.

2 ::.

°"'Yl.C}

�

+ Y-u..+(

d.e-,

�(t>V't�Tr/lf-) � --r

2-:-

&-z.4,(

.: �

s-� Ylof:-

�� T" ::. �TI' "== f/z

�c

1-vd·t, Tj -;; 0.5

�G

4t "�" l , � z: ri-s :: � -;:, .@a ,, 26, ..)

-=- �-=- 2

T, T

Y..1 (:t \ .,,� �d.Lc

T "2-

•

Cc..J-=:, =. 7

To :::: 5 T, -=- :2. 7Y

•

�

;J_

5.· -

w��i5

•

(

{.,u I::

t.<J({z_ �·�

"!:·�.+

t>"177

Tb =-)Vhf,= -I 44-

:..�

------�

(a) For :q (t) + 1·2(t) to he periodic we need some number T such that x1(t + T) + l'2(t + T) = l'1(t) + 1·2(t.) for all t. This can only be true if 1·1 (t+T) = 1·1 (t) and 1·2(t.+T) = 1·2(t)i which can only be true if T = A·1T1 and T = A·2T2 (T is an integer multiple of hoth the periods). So we need there to he some integers l· 1 and k,2 such that A·1T1 = �·2T2 � � = (b) Put in its most reduced form;; by canceling any common terms in the numerator and denominator;

i7·

then To = nT2 = mTr.

1 f Problem 2. I 4

I j

(a) >> syms t >> xa:::::5*exp(-t/2); >> ezplot(xa), grid (c) >> symst >> xc=5*exp(t/2); >> ezplot(xc),grid

(e)

>> symst >> xe=5*(1-exp(-2*t)); >> ezplot(xe), grid (g) >> symst >> xg=5*exp{-2*t)*2*sin(2*t); >> ezplot(xg),grid

(b) >> syms t >> xb=5*exp(-2*t); >> ezplot(xb),grid

(d) >> syms t >> xd=S*(l-exp(-t/2)); >> ezplot(xd), grid (f) >> symst >> xf=5*2*sin(2*t); >> ezplot(xf),grid (h) >>syms t >> xhz:c5*exp(-0.5*t)*2*sin(2*t); >> ezplot(xh),grid

-l

l. l

--- -

---

---------1

Problem 2.15 (b)

(a) t

cos(O+(,) = Re{eJ(o+;>} =

sin(O+(,) = Im{e1<o+;l} = lm{e.fSel'}

Re{e19e1;}

= Re{(cos8+ jsinB)(cos(, + jsin(,)}

= Im{(cosB+ jsin8)(cos(, + jsinf

= Re{cos8cos(, + jsin8cos(,

= Im{(cosBcos(,+ jsinBcos,p

+ jcosBsin(,-sinBsin(,}

+ j cos sin t/J - sine sin t/J}

= cos Bsin (, + sin B cos tp

= cosBcos(,-sinBsin(, (c)

cosBcos(,

. eii+e-1;} {eJ(o+;)+e1(0-;)} = Re ---2 2

= Re { e19

= Re{eJ(o+;) + e1(o-;)} = cos(B+tp) + cos(B-(,) 2

Ptev BLt rY\ · ?{ ')

�,

I fu

X(_-t-, =- � � ( ;;t t)-t'1(

c-n = � (e8

•

-i

"f-Jl� e�2t �(�+J�)e-a2,t

= =

/p 2t - "2r::"" e �7--tJ �k o - e

.L ? 1-,

�(zt)

\/lb'/ti,,' t�l 2

�-t

t»:

e�2\_ \(io1 �:{f�L tit 2.

t�) =-D, 3� Y-ud. (

t V?;. �

= -f o , � 2.. ya

,;-; -J-0,32. }Zt

� (f-) ::

�I

-==

x c+) ::

:z.f.

\ltD'

�l:l. t-o, �7..)

\[io1 � ( "t. -

� 5./0, � (

.� r� 1

fJ-0,32 -a Z-t, e. e - it ?..t--o. 3ZJ

c) 0

1> 2

rio:

'J-..t - ie . 31°\

--------------

•

_..... a

...

'Prob[e'N\ �,[,� ,(Co11t/Y1ue.:!)

(b)

4- � �

=- 4 &r� (4rrt)

1LCt')

ft,

=�ce_,

d,'frT-t:

-Jt1-rr-t) 1-3 (e14TTt-�-o4TT-t) 2-J

2...

=(i-��J) e":" -1--(_;i1-3{Jj

c� - 12-f) = '{c�,-,_� 3

·=::

· 1cC-()

=- { e -=-

(4-TT-t)

1·

- do ,fa4

e�O,�

�?_

� t.frr t

e-Q"'l-lT-f:.

/tc:;,.'

+- 5

}(J ,ILf.l - j-4-rT 6

e e

e �6:rr t:-O·h4) +e 'J-.( :i...

R J : = � /t; ,' t--34.J '·HTt: ._ o- b'{.,)

•

'Xl-f') ::_ -5 �(41T'rf:-O,Co4 h({d) b'\

S C&4 { 4-rr i: - 3�,. �'7 °)

'f>Ro°tsL6W\

J: (

.·. 5;

a,-t-b) �2(t-e.) d,t

•

,:,c �

� {

-fJ

�?._(� -�\�(t-b/a_)

(a:-t-hJ�'-(-t-- c.) J-1: =-±-- �{%..-C0 5;c-1:-!r'cJJ-1: -�

-= it_

�2

?Ro B l6'M ':? , I 9

c�-�')

o:,

�Ct)-=

to..) •

�f1::--b/4)�'-{-t-c"'i

-�

J ( at - b) = 'b ( cJ. ( i: - 6 /�)

-QO

l"7

�J

j x(r)�(ns)-.£fr-sl1Jr

-oa % ( l) s (t:-u_) ::: A{_t<. ) � (r--R) �oa /1 , -r �

: · � l + J -== m:Jy..fc .,.sJ

Jr - 1lC "'';,.�A1t -sJ cl 1:'

- t;)a

�l+')-=: %(-5) -

l 6�

j (t\

=�

'i.{_t;)

��-t 2

X(-t:Je.

-:J>

�6 (;,>t -3) dt

•

t _J L-1 + •

�

--•

-·<

_,.

.....

(c) Recall the rules about integrating delta functions: t�(t) is nonzero only at t = 0. so r(t)6(t) = x(0)6(t). and '5(t.)dt = 1, so .r(t)o(t)dt = :r(0)6(t)dt = x(O) 00 6(t)dt = J'(O). \Ve can time-shift the delta function: J(t- to) is nonzero only at t = to, so J'(t)'5(t-to) = :r(to)o(t - to) and :r(t)'5(t- to)dt = �·(to).

J:.:

1:_:

J�

1�

1::

1� cos(2t)'5(t)dt = cos(2 · 0) 1:: J(t)dt = 1.

ii) o(t -

f) is a time-shifted version of <>(t)i and is nonzero only at t = f

: ,:;

-,,

So:

oo. sin(2 · .ii. )o(t - �)dt sin(2t)'5(t - -)dt = -I J-ex· 4 -C\: 4 - sin(f)J"' J(t-f)dt=sin(t)=l ... -00 ... -

------------------------------------· • u.l U-i.i) j�L-;. d .;1 ,1'1

c) ;; (t - %)

=-J" '�c��1 s(t-TT/&Jcu-

-�

�

=-�(:Z) Jb(t-717t,)Ji

�%; = o.scc

-cl'

- -VO

5�{ t-%)1 £-Ct - �)dt

�

.,tJ

(Lil')

-: : �r�lf s� (t -7%) �(-1i _".::f>

-�

•

s·

- �

==-

A--{(-1X-)Ut-rg)Jf

�

�(f-•V2.)lt -::: iJ,7o7 �

{:)a

(tr)

;i�Jdt jA-w-(t-'%.)&{i.(t-,v,JL-t - '['"-::I

,.,.

� (%-1TJ,,\ S[:z {t-.IT7,)1 J t" i h..{%) ::-a. ,15

1 11

+ -

•

l � �

(a)u.(2t

+ G) = u(t + 3)

-3-2-1

I _. 11

1 2 3

�

...

-.--

(C.)

t� + ., )

i,c[:-B(t-2..")]

�

� L{, ( -t-+

�c� l*-

r,t (

2 1- �) ")

u_Lt+s)

t-,--3"'

-� I

cc1 > u

c� -YL,) �

L\ tc-o/2-)J u L-t..-3/2.')

[

= u.

< t -312-)

-41

j u.(-t iL-2.)

•

I: -+-

U(-3-t-+4') =

l :

.....

•.•

•

. '. - ; : ( 1 - l L

u. (u...(-t) -

I - U

. .

• [ r

� _ cb) !

. .

• ,

t .

u. c- t: �

c\ -

-= t

l . . I t . ' -· l

I .· .

J :- · 1·- ·•l

•

� 'I

u. ( - t + 2) ::: u. [-1 (t - 2'}] -l

-t,{

::: Ct-1., c 1

•

Ct-2")

�c tL[-, (t.-2-')]

u (t-2) 1

��--+-��---��t_

�

I I

•

...

�·1 I- ·• :I !. :I :i •

!-I '� '

I .l , _,

,- .

l �

I - U.. ( -l -2.';

Lt-2.'>

Cd') (-l-2}u..La-t) -=-

�

....

.!.._�

,,..

. -1: u, (;-t) - � . •

�

...

I .

• ·1

«, (t')

l --.

;

r-�------+----Tt

_ • '. - • •

•

- · ·\· 1 ;i ·

t ')

)

-t \ ,

.. . I •

•••

-l

t l�.; 1

• • j

•

: - · -� l ..

-1

--<

�

x(2t - 4) = 4j.(2t - 2)u(2t - 2) - (2t - 4)u(2t - 4) - u(2t - 6) - (2t - 8)u(2t - 8) - (2t - 9)u(2t - 9) = 4i(2t -2)u(t -1) - (2t -4)u(t -2) - u(t - 3) - (2t - B)u(t -4) - (2t -9)u(t - 4.5)1 (()...\

1:(t)

9 8

·(t - 4) -1-----,

9 8 7 6 5 4 3 2 1

-2-1

1 2 3 4 5 1 2 3 4 5 6

'(b)

.., I

8 9

.1·(2t - 4) D 8

·�

6 5 4

·�

3 ....

2 :3 4

•

�

- - ;

-e -,

�ct".> ::: � u..(t:+

-•

J - u. (t) -F 3 ll(t- 3) - 5 u(t-(o)

- l

(?t._,')

X,[t') I

I I

-3

1 I

- I

�

:e J

I I

•

. I

-·

. 1 (b') x{�-t.-0) ::: 3 u..( 3t-�+ �')- u.(�-t-'7')+ 3_u..(-a t-l,-3) ' - 5 u. (at: -G,-�) '

zz: 3 Ll.

(*. -"3)- U.( "3t -4,')f 3 (( ( 3t-q) -5 U ( "3-t-f ZJ

-= 3 U..(t-1)-U.(t-2') f3U.(t-3)-5U.(t-4) 'X.( :,t-4>) � I

I ,

-=--�-+--+�-+---f-----t-�+----+-�4-t---�, -2. -{ 2. '"C

•

�

•

H-i _1I

I - ,_ -

. .... f'r?,C)S

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•

t . � -_

2, "-�

-�

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j .....

- ....

+ - •

r- -'-

: !

•I

-··

- -

-1· '

, I

-j

b·

.:=- - -·I-=R \ I I

-1 - II

' I

l � ' 2

">fa.-> -r j

t 1,;,{.._

a.� d a...mp [, +1{.d. e b-ccn.s £ y L ..L.

�

2-..

•

4 I

� - -r 2... •

- !

12...

j j I L lA)

th)

�,/t) • lt1,t.ke)-fli-1) tA.lt-J) .t<D

X, /&)�t)

+Zlt.-Z)IA.l�--l.) _

b-'Z</ J 'I,�) .::�,,,

J<.i�z.J 7., l:t )= ZZ-.l.J,,Z. ->'I � Jf-�z ..... z-<J J � ,JJJ = J/-l .t. +2z:-JJ.::.t:J.,,.. te) �tzJ .. � :r,£.ihh.7;.} :t,ci�zJi.)

=$,

•-M

f,11�

�r,::_ c ffs L-6 ti\

(4.)

� . ;z (.p

x. 1 ( t-)

r '

I --

-::: 3 [t

-3

L(_ (t.\

rft-2

(tFQ�J'll,(-t) <o (o ci: c 1')) 'X..1 (-t"') -=- st

-E_tl) Ll(t -i) J

Gt(t-2) -�-'3 ')'u(t-3"')]

!=+f-

i-:

. W-,< t �

2.) ; 'k.. 1 Ct-') -= "3 t - 3 t: + 3 -= 3 V (� <i: -c 3 ) °XL (_f-) =: � �+;3 -:3t. +{p (-t2.p )

1-, l+, -== p:t-�-r;,!f'-3rf:+_);5 +3-:f::-g-=.

-?J-t+9 0 �

lb) 't2-l-t J i�

periodic.. wi t:L, � .c,.a

1l:z. (-t ') -=-

e: X,

YI =.-oo

=- £.

+n)

3[-t +-1/-.YT)u.(t-+4->i )-{t-f +4-�k{t-t+4n�

YI --De

(t -1-

= "1--

_ "3 [(t- -.2 +4-YJ) u(t-z+-411J-{t-:w1 "h) iA/l.--3t4�

'P�eL;&M 2 9':J I

;z.zt) * J;.[r,C:rtt>J] :, 51s_ttl-=- T:r[T. rtltJJ] )# /:t-)"' Tz rr. r,: u-,JJ "..J, hJ Ts tr. [1! Lt-JJj + )s fl'16>] Ct,� �Ltl= �fTz.fT,r.%(t>]Jj.,.;f{n;n-�.H + i; CT,rtu,J] (A.)

\�l�L\5C\(t,]]J

j�(t') � 'a 4Ct') -=- LL[14CT5

lC'\

lj5 (� ) � 1r,

� (t) = 1

•

Gjlt-f)JJl

er� c 1(l+_)JJ

'd 3 {-f) -/- 1j Lf (i:)

"'r

d 5 (-t-)

:: '· ( )

ljfti) � I� LT� [1 � LL(-t) 1J] + T�[T4l_T5 [X-(-t')]J - 1 · f- Ti c._!'='l X.. C-r') ]]

Ld)

�C-t."'1 =- �:il-t ��4(t)t<-�5(��

d (-t') ==13LT4 �i{%<:t')]1Jx Til4 Lfs[X- C-t-�1� )(.li C-r�[x (.+)1J -Pre..r.,\"3L5M � v.vJ l-t o

-':" Tz,

�{+_) �

� ('t) =

,28

.(x: {+')- T4[� (t1]}

Crv1 l {)

-l-

(X, {f ')=-1.

T� [-r:z ['x.it\-T4 � (·f'lj ii', tx.Lfll}

?tw�L6M �.2-°J_ 'IV\ L -1:1

-= T�t_,c (-1:) -T:, [�

ct-:ilS- T'+ [� l+)]j

\jl�')= T1L_W\(+1]

d L-{) � 1,11

·lv:

(tl -:'I -T3 �JL·FJJ} T4

�Hl�

. I

•

;

J-

D t 0

iP 0

� 2.

*)=

I::.

-7..

-I

-1

� \ - t: - I) ::: \) 1 ['L.(tt I)

f::..

.. .

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+ I

--+

, I

•I I

1 .... I

�

I t

•

�2

1 �

{b)ti)�

Lil)

a=� jl«-1l

£:l) J

,.;! ��

liii) CAAA8�

•

lilr) ��bk (Al')�

(.-i)

,J,,. ,

itt� ::f �}

�,.s-1

r,+ii.

nd � ...

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l�

Ld) (,,t)

VV\.e Moy� ( 2�

•

. �v

a-ti i;-aJues

o.P t.(t-)4.-1.

ci-i) nti 111u-eYh�l-e: (-t> =-t-h)'(' t1..l( t(f-; -,2.._, (-iu) tac.z csa.. [ \.iv-') S-cc,_ b ! €. : 'O � � l t \. '=-lf• (_u-) -t1 'me i f\ v-tt. i,-·}a Yl t ( U---i� "'Yi st: J 1'n-eC{_r : � l+1 = 1- +o Y- a: t f ')( l-t) 7 2

•

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