8
},2 E [E,J VunLg") 7 d P U{^o, U/r^e V"2 e = 7( E tx;l = nA ( Flx, J' ^t n n l{ ,? 1 -4 trarT.Xpj + Z. + hn[X j. )=, l-=l v j#[ ' q= EIYn) = ,tt ) xp) n : z i=f n : VI = n6'+ )cn-t)p6 P l<el P>o'L t-o.oZ P(Fn= 0.t{ t f l / f,<2,<e+ne { n = Swo DTrSn I tlfr I - //l Van(E^1 n- ' O-+ n+ O = n 6t* >(n1) Pa' D1. rr I (17-- e ,.Lt / > l _> o. t J . T/"'e'{:'y. w.o*lc /gr-' 4 LT" ,tar-,,-ha.. ,bL/, 'Tt.>7

VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

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Page 1: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

} , 2 E [E ,J

VunLg")

7 d P U{^o,U/r^e

V"2e =

7 (

E tx ; l = nA ( F lx , J '^ t n nl { , ? 1 - 4trarT.Xpj + Z. + hn[X j.

)=, l-=lv j # [

' q= EIYn) = , t t )

x p )

n: zi = fn

:

VI

= n 6 ' + ) c n - t ) p 6

P l < e l

P > o ' L t - o . o ZP ( F n = 0 . t {t f l

/f,<2,<e+ne

{ n = Swo

D T r S nI t l f rI- / / l Van(E^1

n - € '

O-+ n+ O

= n 6 t * > ( n 1 ) P a '

D 1 . € r rI (17-- e ,.Lt / > €l _> o.t J .

T/"'e'{:'y. w.o*lc /gr-' 4 LT" ,tar-,,-ha.. ,bL/,' T t . > 7

Page 2: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

t A f o . o 2 . l

u LG- l= o . r r J - - ) + = t fSA+ ) -t_t_

d T

& -.1.?s '+ n= ts>l2 - D

7.7) Trtt l errtr Et* = '#rn, , Xi r] &:t etrtr trt

i tA ttv'u'4t++lr*i d

It*a c/,r,*{f mcu,*tnV +p 7 )' '- )

lLt,",z+1 s (ffi" )'""

lh,x1 "c tT , f L Ep" >+l = e f a.=_/ =, oot\Ejt r

Page 3: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

t^fk tf* h (o ,t )t .

dir*r (x ) 3 P(*c+) <-) t*,V qu,valp-rt e",e',4$

t b L a W f t r c , V

Ktr*)<x ̂ fuA z<6<Ttx+ G - x < a

Itxrc)sx;= ft+rk)ok)60-z7o

- J - t= T ( t r - x )= x

F*( t ) Lx) =

o{rztn b,t*w

j*t t/s

xrc)

f,wrtn f*

t f L

1 t$)

Uso

l t r < x

r r o < T . . t nf - X r f o < T < t -

f:fic r> ok + r ' rr,')o{ r12 f x- t ' =yt t -x

$

f 1 x < o \

f o c r ) e ) = ll z o < t < l

+

i l , > ct\ (/^;-f*^ ^^/ d,', n7d ,3

[*'*Lt4"{' 4 t>*-o , ir t) y*s;l{' -tP La =z

T" , <7 - l t z

D T

F( x Ltn,) < *) =

=

do

t L

q,W

f'xct,t

Page 4: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

l / t Ii o l d r : ,

,l

o < 6

, b t f h * , t t ) =

t (f*,, =

L'E [* rc\K(t *a, lz =rJ tr(r ) lr+ . (t L x(r )

: f lj o

X ( f * a

h , . t

= f ( r - A ) ( a+ f ^ ' ( r_b(

t

^ , - 4 1s ? ? 7 + 3

Cx'[xr* r

= f,' f(+- k) A (t+ 6 - l,J r{'L

trGh) il (^- I-t,{a

4

-A ) l ul - k + o ) / k

V r + * o ! = E [ K r c ) y f t * 4 ]-E [xc+L) Et*r+*4]

d

tg2

Page 5: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

?,$ ,"b I

V ̂n,(s,,*vaht,,

= D t - r \ I\ { r t 1 ) ( - t ) =

tb aw atf-til{

9 ,? ^ owfor^j

f*/rzti

cftu,ut t

b*fl>n . 1? 0

P\ Ave" 6 & r*;arc

l Y \ H k ) ' I P * u j , )

Cn(+, , t ' ) =

t ^{,e* tr_t, L-w

rc,> + Hff-))

, )

0 e (*7, rr) w

P (Xru )Yc +,) >o)I'

= f t - * - f l < t n< + , < - o r Z , )\ rz - VJ

' TArl)

k - trw +1]',)

(* tr+ fi, aw fltx.t )x(rz) ;r)

= F f t , 4 - * - ? *t* l>r t,I

= f ( €-tw€ g <qat-Y ) -r [r(T*t,r 0<-t;

= t * - E + j

( t )

bhfa^el

Page 6: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

7 - L w + t ^ wI\ l l ultt

$ilfli l -ilTI

( t , ) K(+ ") >n)

A . - t vw t tw

F t X{

- ) tt u - l lililI

ry 7 . '

Cufu, t , )

Page 7: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

f.16 rA) rTty^]V* tI, l

? e \-Ccr, ) X"J : (In) efX^]

) (c'c^ >J.v*[x,] + eix"]la - ' / , , . \ \ )c - ( n ) Van 1" " ) > c , ( n )

rb)

= VwrIct^lx"] =EL6r*;] = c'(n)Elx, 'J

' !n* , ' t ' J<xt , C(.r .}Xn*, . t r l

f f 1 ^ . 7 q

f I c'") X,,

rEr{ y n , L n ,

f^'f-x ,,

tF,f-I xn+r

( X , , X t ) =

r x n + t ( x t x ' > t 2 r t ^ \ \- _c * t ' f t ) ' f c c n )

) - F *^ , x6 . . t ( k , f f i )t t ' c ( t r * t ) 1 , t . c c n ) > t >

X - f r x t \u fXn [ Cr i l )a &n*, ( x + - I - + E I v t x a - \'

c ( n+11 i l l \ " / n * , ( 6 rGD tc ( n ) , c ( r r + t ) < D,Y

Page 8: VunLg) Z. + hn[X j. xp) VIssli.ee.washington.edu/classes/ee505/hw/hw4_solns.pdf · t^fk tf* h (o ,t ) dir*r t.(x ) 3 P(*c+)

= r[y,., y^, ] - e Lr,,J ety",l: C Ln,1 CC ny7 e I,Xr., Xn,J

rc) q { .n , , A , )

f c , (m) Xn X-]rn ) f lXn x_ ]( t n ) R * ( n , , v ^ )

CL

C