!

" # $%

& ' "() *+ ,- .

/ 0

! !

$% 1 " ' 2 3 1

" ' 4556

7

) " 2 " " 8 0 0 ' ' 2" 3 % 1"

' " 2 " 9 " 9 " " ' ' ) 0

:; 8 < 8 % " '

" % ?" % 8" " ,#" '

@' " 2

" 2 9

# " " 2" 2 '" 2 ' 2 09 00 ' %@ 0 " 2 9 %" " %"

0"

0A " >" %' '" '

3 @ 2 2

0 3"

8 # 2 0 #

B" (

92 0 ' ' " " 9"

"

" 9 0 = " " 9"

" 9 0

' "

% % 9 ' =

% 3 #" '

" 9 : ; " " ! ' = ) 00 2" " 09 9 %

" % % ' 1!

2" " 9 %

= "

'

" % # " = =" 2 C3

" = " 0

3 = '

#

' " =

= 2

9"

" %" %

%1 # ="

,

) " 3

= # %" 9

2 %

' 3 ' 2 = %

' % =

= '

"

' =

! # ) 0

=

9= %" " 2

! 3 0

' %%

3 = 02

17 "

= %0

'

# " " =

" ' =

=" ! # "

' #

0D )

3=

# " 2 % " 1 2

' # %" 2 ! ) 9 = 7, '

' " ' % ) !%" 9 =

" #

% " '" 3" = "

% ' =

0 E

0 # 0

" # 02 0% %

F = 0 '" " '="

"

0 02" 0 ' ' 0 '

=

" 0 " ' 02 0% ' 0 % # (D%" ' =

02 0% D0 0

'" ' 0 % # 0 0 7'" 0 0 %D 0 02 0 D0 # 3 "

'" '= 0 # # ' 02 : ' ;D0 :" % " ! = ;" E 0D 0" =

" = 02 F # 0 %

0 0 0 0 # % 0 DE 1! 0 0 # % ' 02 D

0 F =" '

! # 0 0" D D # ' # "

=" D00

D # D

# ' " 0# 0 C3' # =

0 = 0

%

# 0 D # ' 0 = 0

D 0 D 0 D " 0 "

D = 0 E 0" % 0 % # = " D 3

0 =

!

'" 0 0 = 3 # 0 D

0 0= % # 0 D

% 0 '

'

3 ' # 0 D

%

D % 0 0 0 %' # = = G 0 =

" 0 D

F )7)

00 D !

0 " D % ' 0 2 =

02 0 0 # 0 ! # D= % " D00 '

'

' :# " *7;7

'" D 3 0 D

=ED

D # !

'"0 0 3 0 # 0 : D

0 = ; 0 !" =# 0 ' D0 D0D 3' 0 0

' ' 3

D0 ' 0 70 " D0 0% % "

=# 0 %

=7, 0 " 0 0 D # 0 0 0' 0

0 0 % ! # #@ 0 # # 0! 0 02

D

=7, # 0 % D0 E 0 # 0

'" 0 0 D ' 02 0 % 0 . =

/ # %D % # 0 0 %"

% # 0 02 =" %

"

#

#

$

%

HH H4 ' % HI *

4H 44 ) 4I 3 " '

4IH ' 3 46 3 46H 3 464 3 # 3 4L 4LH ) 4L4 9 4LI % 4L6 4M ' ( 4MH ,%

4M4 % #

4J

4JH 9

=

HIJ

KKHIHLHMHJ454H44444I46I5I5IIIMIJ

N+) >)+),8

&

IH I4 # I4H

I44 I4I II ! 8IIH # II4 7 8 # III

II6 IIL I6

' (

'#

6H64646MLIL6LLLMLJMHMIM6

)*

6H ML64 7 = MM64H MO6I ? = J566 = J666H J6664 ! JM66I = O56L = O46M % = O66MH % OL6M4 % = = OM6MI % = K56J < = KL6O & = KO6K H556H5 > # H546H5H ? = # H546H54 = # H566HH H5J6HHH = H5J6HH4 = HH56H4 = HH56H4H %" % =

= HH56H44 = HHH6H4I # = HH4

N+) >)+),8

6HI HHM

*

(

LH L4

= L4H )

L44 % # % L4I )

G8 L46 %

G8 L4L )

L4M )

% L4J LI

! LIH )

LI4 LII

3 LI6 , '

L6

() L6H ,

G8 LL

)

MH M4 M4H # M44 7 MI % M6 M6H M64 7 ' M6I % ML = MM ' MMH MM4 MMI % MJ 7, MJH MJ4

MJI 1 # MJ6 1 ! % MJL )1

##+

HHJHHKHHKH45H4IH46H4OHIHHIIHIIHIIHILHIOH6IHL6HLLHLO

#)#

HMHHMIHMIHM6HMLHMJHMOHJ5HJIHJMHJOHO5HO4HOIHO6HO6HOLHOOHKHHKI

N+) >)+),8

MO HKL

+ ,

JH ' J4 8 # J4H 7, '= J44 = !I J4I % % J46 J4L 2 $,6H ? =" [pp+q] (t)" p = 1, 3, 5" q = 2 ' %

5 ' H 64 ? =" (t)" %

6I % # =" %

66 )2 = 6L = ! 6M = 6J = # 6O % = ;

% " ; % 6K , : ;

% = 6H5 = = ' % # 6HH :;"% :;" :0; '

% :1; =

% 6H4 , : ;

% :2;" % :; ' :0; = 6HI ? = :; ' % :1; =

% [2, 0,4, 0,8, 1,4, 2] = 0,2:; ' = 0,7 :0; 6H6 ' = 1

6HL = ' =% = 0,3 ' = 0,8 6HM = %=

2 0 " %' 6HJ = :% ; =2 0 " % '

6HO = :; ' % :1;

% 6HK # = : = 0,328; # 9++," G8 ' 645 % 9 # %

C1

MOJHJIJJJOOIO6OMK5K6H5HH54H5IH5LH5KH5KHH5HHHHHIHHL

LH 7 % = H44

N+) ) 7>$,L4 7 % :; ' % % K = 4 : = 0,5; LI 7 (, x , ) ' # ! x1 0,25x2 = 0 L6 ,

LL )2 ( #= ; " ; LM ? ! 9

LJ :; '

:=1; *7 :BWi = 1 (" Pi = 0,3 ? ; LO

%% 1 *7 BWi = 1 ( ' Pi = 0,3 ? LK >! =

%% 1

*7 :BWi = 1 (" Pi = 0,3 ? ; LH5 >! =

%% 1

?= :BWi = 40 R(" Pi = 1,6 ? ; LHH

! = *7=M6- :BWi = 10 (" Pi = 3 ? ; LH4

%% LHI # yI [n] ' xI [n k] #

k *7=-R Pi =9 ? ' # 0 LH6

: ; ' =

G8 *7=-R BWi =6 ( ' Pi = 3 ? LHL ), Pi = 0 ? :2; 'Pi = 9 ? :0; LHM , Pi = 0 ? :2=

; ' Pi = 9 ? :0; LHJ % 3 '

G8

() : Vgs0 = 0 0, j = k.

1 * k = 1 k = 0, 1, . . . , n (

46 3

HK

1 1 # 6 /" {Pn(x)}" [1, 1] # w(x) = 1" %

P0 (x) = 1,P1 (x) = x,Pk+1 (x) =

(2k + 1)xPk (x) kPk1 (x).k+1

6 ==B" [1, 1] 1

# w(x) = 1x {Tn(x)} '

2

T0 (x) = 1,T1 (x) = x,Tk+1 (x) = 2xTk (x) Tk1 (x),

k = 2, 3, . . . ;

" # 3" 3Tk (x) = cos (k arc cos x),

k = 1, 2, . . . .

# 3 " =

:

0'=0F; ! 3 =36 >" {Hn(x)}" (, ) w(x) = ex "2

H0 (x) = 1,H1 (x) = 2x,Hk+1 (x) = 2xHk (x) 2kHk1 (x),

k = 1, 2 . . . .

6 " [, ] = # w(x) = 1 {n (x)} ' 30 (x) = 1,1k (x) = cos kx,1n+k (x) = sin kx,

k = 1, 2, . . . , n,k = 1, 2, . . . , n.

45

4 # f (x) [, ]" 3 =

9 Sn (x) =2n1k=0 ak k (x) ) " n " Sn (x)

7 f (x)"

. " 1 # sinc (x k)

(, )

w(x) = 1"

sinc (x k) =

sin ((x k))(x k)

k Z.

V 1 # !

) 3 # % 3 # 3 " f (x)" ' % [a, b]" " 3

= max |f (x) P (x)| ,x[a,b]

2 2% L ( " =3 9 3 3 3 3

%

" L2 " 2 =

3 " 0 : 2 != % 3 ; + " L " 2 ! 3 # " 9 9"

% 3 #

# " 3 3 # G : 44;" 2 " "

00" = " 2 "

" 3 0'0F" :P0'O4Q ' P%JLQ;

$' 9.A: 1 x1 , x2,. . . , xn

2.%. n Tn (x) xj = cos 2j1 1 f (x) 2n 3445 Pn (x)

& n Pn (xk ) = f (xk ) 6limn Pn (x) = f (x) ( 3445

4L

4H

) # 2 3

0'0F ' 3 3" " n" 2

% 3 # ) = POKQ N !

0'0F" cj j = 1, . . . , N " 2cj =

N2 f (xk )Tj1 (xk ),Nk=1

2 # 3

f (x)

N

k=0

1ck Tk1 (x) c12

:4L;

3 x N TN (x) ) =

2 " " 1 3

N # f (x) 3 N != " " % 46" 3 :4L; % 3 # f (x)) 00 2 ! ck "

" % 2" 3 m N 9 "

9 9 cm+1 Tm (x)" # m + 1 % 3

# % [1, 1] 0 2 3 0'0F ' 3 13 # " 2 2

"

8 9 1 2 #2 8 3 #:; ' 2 % : ;

" 2 #

% =

:' ' % 29 2

; 0 9 =

0 # " # 9

%

" # #

44

4

) HK6K" ) 0 .

( /" # P06K"0KOQ ! # " 2 % "

2 1 " "

" 1 % ( +'2 P +'24OQ

$* 9 : () f (x) &( & max7

T = /max

" # :;

f (x) =f (kT ) sinc(x/T k),:4M;kZ

# " f (x)" 2=

" f (kT ) 0 3 '

" " # 3 max /T "

(

P154Q" 0

= - 8 9 : 1 % +D 'G;" 2 ( # 0

2 % # %) 8

8 $#& 9 /: () f (x)

3%5 n + 1

Pn Pn

x0 , x1 , . . . , xn

.

Pn (xk ) = f (xk ), k = 1, 2, . . . , n. ' $(

) *(+,*(+-./ 0 "! , "!-1/ " 2 3 ,34/ ,5/

! # "

6 7 8 9 '(" * 9 : ,:-/ " ; 8 .

:IK;

f () =

0,0,

;

I4 #

6L

$ :I6; ' 3 #

' # #

&

Pa =

1

fs2

0,

fs1ln2 , > fs /2;

max fs /2.

:IH5;

2 3 # '

R # R=

fs,2

3

& #= R

Pa =

1 R (1 ln R) , R < 1;0,R 1.

$ :IM; ' :IO;

Sx (f ) =

?

1,ln2f

f ;

Sx (f ) = R ln (2Rf ),

f

45

1.2R

2 # 3 " " = 3 c" 1 e c,cF () = 1 e c ,f () =

0; 0.

:IHH;:IH4;

2" :I6;" fs)i2c

fs

Pa = e 2c

fs2c

,

)i # 3

)i (x) =

x

etdt.t

:IHI;

6M

I 9

c

2 % 3 ' R % "R=

fs,2c

3

& 2Pa = eR R)i (R).

" :IM; ' :IHH;1Sx (f ) = )i2c

f= R)i (2Rf ).c

&$#& 2!

) % 1 2 = % ) #" O O555 8 % :%" ' = ;" 2 0

! " =

% ' %

: [0, 11025] (;

7 IH " 1

# 2 0

" 2 # ' #

# #

!

# 9" 2

" ' # ! $= >44I ' 7 IH %" % # 1" 2 # ' " " " 2 2" ! "

#

8

% =

# '

2 7 I4" M

" x[m]" M % % :M > 100; 0

I4 #

0

0

10

10

20

20

30

30

40

40

0

2

468Frecuencia (kHz)

10

0

BC

2

6J

468Frecuencia (kHz)

10

BC

. 8 % ; " ; !

# 9

! ! 1 # c = /M: 9 77 =77 ; ' # M " 9 " xs [n]" fs = 1 8 x[m]" fs = M " % # 0

9 xs [n]) !

% #= ' " nq [n]" ' % 2% % ! b ' =

> ' x2 "

! P'O6Qn2 = x2 10

76b10

,

:IH6;

2 1 8 :=

0

%;" %

! n2 = x2 10.:IHL;7 " 77 "

77 " 2

=% 2 2

" #" x[m]" " x [m]"

126b10

$

P (|X| > xsc ) = 5 105 ! 3 ! % x = 4x > D 9 P (|X| > x ) = 5 105 x = 7

x

6O

I 9

nq [n]

x[m]-

7

?1c = /M

xf [m]-

M

-

77

-

? xq [n]xs [n]

(L)

-

77

(LM )

x [m]-

x [m]-

. &$ )2

&$$# "

) " 2 " x[m]" = ' # = " R" 2 3 # " fc" ' # +'2" fs /2@ "R=

fs.2fc

:IHM;

) ! 1 :f; > ' =# fc = 2R" x[m] % ! :2 # 2 R 1;" ' !" =9 " xq [n]" 2

9 : ' ; !

% ) 0 N = 100

LM : L = 1000 ' M = 100; % "R [1, 32] ' " b [2, 16] ) 7 II = % Rb

' 9 9

Rb 2 % 1 1 "

b = 8 " 2

# R = 2,2@ 00 9 0

"

! ) 2" R b

" 1 ' : 0,5 ' 1 ?;" 2 = R b % 1 % ' 1 * ' 0

9 s

I4 #

6K

8

Factor de sobremuestreo

76

Spline

543

Sinc214

6

8

10N de bits

12

14

16

. && ;;" 6?/

0,603

0,377

0,445

0,3843

1,412

(

0,602

4 % 4 %

'' # %

C1

: = 0,289;

HHM

6 =

! ,

) 0 % # " ="2 #

= "

0

2 = # , ' " 2 =

L2 ' 2 = " ' 3 2 # 2 ! 2 ) ! " " 2 = = # ' %0 2" "

' " =

' 2 %

" 0

#

" 2" % " 0

# = 9$ # " "

=

% = ! : 2 2;" 2 = ' ! ) 0

3 2 0 '

= 2" ! = 2 %" % C"

0 '

) 9 0 3 2

%" % ' =

" " 2

%

0

% = %" % ' " :'

;

% 2

2 = )

0

% 1 "

# "

%

:' ; !"

0 = = # "

" " # "

"

)3

" 1 % =% ! " 2 9 "

.

8 %" K " 9 2 =

#" ' ' % K=

f (2) (x).[1 + (f (1) (x))2 ]3/2

) " " x = 0" % :LL; ' :LM; 2 ' 15/4" % " '

% K " # # %

" % 2 # % ' 2

% ' 2 % %

%" 2 2 !

% % ) # 2 ' |x| '2 % " 1 P>\0JIQ"

hyp(x, ) = x2 + ,% K = 1/

' %

, " 0 # P85HQlch(x, ) =

1ln (cosh (x)),

:LJ;

' % K = "

%

% :G8 1. @8

, ;) # % = 0 #" 2

(x, )" 3

(x, ) =

x2

3

|x|32 , |x| ;|x| 3 ,|x| > ;

:LO;

H44

L

=

3=0=1=2=4

2.5

2

1.5

1

0.5

03

2

1

0

1

2

3

. *# 7 % =" (x, )" % %

%

' % K = 2/ ) 7 LH # %

%

8 % # :LO; %

(1)

(x, ) =

'

x|x|2 , |x| ;sgn (x),|x| > ;

(2)

(x, ) =

2x

2

0

2|x|2 ,

|x| ;|x|, > .

:LK;:LH5;

8 7 L4 # %

% %

% K = 4 %2 % hyp(x, )" lch(x, ) ' (x, ) " " % # % 2 ftan(x, )' fpol(x, ) " 0 % 2 hyp(x, )" lch(x, ) ' (x, ) = ' :' ftan(x, ) ' fpol(x, ); 0

%" 2 0 # # %1 " # (x, ) ' %" 3

:=;" % # 0 2 2

2 % 3 tanh (x) ' sech (x)" 2

: ;

' %

# 3

L4

=

H4I

21.510.501.5

1

0.5

0

0.5

1

1.5

1

0.5

0

0.5

1

1.5

210121.542

.

021.5

1

0.5

0

0.5

1

ftanfpolhyplchspline

1.5

. *$ 7 % :; ' % % K = 4 : = 0,5;

4-

2 % 0 #

" 3

) "

%

= 2 f : RM RN G8(M, P, N )' # f (x) = a + Bx +

P

ci ( i , x i , i ) .

:LHH;

i=1

) " # i=9 ( i, x i , i )" # %" ' %" 2" "

) 7 LI 1 0 # ' = [1 0,25]" = 0 ' = 2/K = 0,5@ 9 # ' %

H46

L

=

3210122

x + x =01 1

1

2 2

0x2

12

2

1

1

0

2

x1

. *& 7 (, x , ) ' # ! x1 0,25x2 = 0

2 # 2" 3 % # (x, ) 2% 3 = 00" 3 # =" (x)" ' #

%" (x, )" 311 (x) = (x, ) + (x 1, ) + (x + 1, ) .22

:LH4;

V # % #

, :? & 1 ; P7 KHQ

#! 4-

0 6" % # =

= # %

:LH4; 3 = ' =3 =" 2 %

G8 :LHH;

L4

=

H4L

) 2 %

: % !

2 % ; " =" !

1 ' %

8 9 0 # 9 3 '

# 2 %

" "E=

(f (xi ) yi )2 + ||Ps f (x)||2 ,

:LHI;

i

@ " 2

% ' 1 @Ps #

8 %

# ' %" 2" =" P % #$ 2

2

# PK5Qyi

||Psm f ||2 =

n

i1 ,...,im

(i1 ,...,im f (x))2 dxRn

i ,...,i = m /xi . . . xi ' m 1 ) " %#

" 2 %

"

% ' ) "

2 % :%

% ; ' ' 2" " %

% 0 # (x, ) ' =" 3 =

G8 " 0

# 1 " 3=

%1

m

1

m

H4M

L

=

# 2 % # = :(2) (x, ) = 0 |x| ;" %

=3 9

" 2

b 0" "f (x) = c1 (x, ) + c2 (x b, ) .

) " 9 % 3 22 8 b/2;(c1 + c2 ) 3 ,4(2b)322822||Ps f || = (c1 + c2 ) 3 + c1 c2 34 ,b/2 b;33 26b2 )8+ c1 c2 4(3b +4, b;(c1 + c22 ) 334

:LH6;

2 "

%

2 b" " " c1 ' c2 @ % %"

9 )H

3 "

2 % !

# " % %

! ! % " " R

= #" ' 7 L6 "

# ||Ps2 f ||2

=R

2fx21

2+2

2fx1 x2

2+

2fx22

2 dx1 dx2 .

%"

=" "f (x) =

2

ci (m1i x1 + m2i x2 + ti , ) .

i=1

9

" t1 = t2 = 0" 2 2 # " ' 2 m11 = 0" m12 = m22 = 1 ' m21 = m ?1? !

#

m1 x1 + m2 x2 = t

!

! "

L4

=

H4Jx2L

x1L

mx1+x2=0

. *' ,

" 9 %

||Ps2 f ||2

=

R+

R+

R

'(2c21 (2) (x2 , ) dx1 dx2'(2c22 (m4 + 2m2 + 1) (2) (mx1 + x2 , ) dx1 dx22c1 c2 (2) (x2 , ) (2) (mx1 + x2 , ) dx1 dx2 ,

9 = R 2 9

" 2 ! 0 7 " ||Ps2 f ||2 =

8L 28(c + c22 (m2 + 1)2 ) + c1 c2 ,3 1m

% d||Ps2 f ||28L 2=(c + c22 (m2 + 1)2 ) 0,d32 1

2

%

=

) !%" % 2

%" 2 %

H4O

L

=

)

= 2 2 ! # 2 : i ' i ;@ 2

:a" B ' ci ;@ ' % ) 1

%" ' %

'

!

0 8

2 ! #

9 " "

' ( # $ % 2 !1 #

" 3=

# 2 ! " 2

# ' % " %% # ' % V " " 9 8 0

P0OMQ" % 00

2 # % " " (x, ) :LO;

# %

" 3 2

=

3 (f : RM NR ) 1 L =

(x[l] , y[l] ) [l][l][l]l = 1, . . . , L" x[l] = [x1 , x2 , . . . , xM ]T % l=9

: M ;@ y[l] = [y1[l] , y2[l] , . . . , yN[l] ]T % l=9

: N ;) ! # i, x i :LHH; di (x) = mi,1 x1 + mi,2 x2 + + mi,M1 xM1 xM + ti ,

di(x) i=9 # % x "

= :LHH; P # y = f (x) = a + Bx +

P

ci (di (x), ) .

i=1

8

%" zf " =

P M 2 ! P # 2 zf = [m1,1 , . . . , m1,M1 , . . . , mP,1 , . . . , mP,M1 , t1 , . . . , tP ]T ;

L4

=

H4K

' " Zc " N (M + 1 + P ) 2 ! #

a1 a2Zc = aN

b1,1b2,1

......

b1,Mb2,M

c1,1c2,1

bN,1

. . . bN,M

cN,1

c1,Pc2,P .

. . . cN,P......

)

#

'

L P

2

[l]

E(Zc , zf ) =ci di (x[l] ), .y a + Bx[l] +

:LHL;

i=1

l=1

% " !1 # " " % zf ( " # 2 # $ % !1 zf " # :LHL; # Zc " 2 %Zc = (AT A)1 AT Y,:LHM;

Y = [y[1] , . . . , y[L] ]T L N

" ' A (M + P + 1) L #

1x[1] 1 [1]A = xM [1]u1 [1]

uP

1[2]x1

[2]

xM[2]u1

[2]

uP

...1[L] . . . x1 [L] . . . xM ,[L] . . . u1

[L]

. . . uP

u[l]i % i=9 # % l=9 " " u[l]i = di (x[l] ), $ % Zc "

% " % "g" ' (" H" :LHL; zf " 2

" s" 2 ! zf " ' 2 % 3s = H1 g.:LHJ;

HI5

L

=

)

% 3 zf " ' 2 #

:'" " ; N = " % N

E(Zc , zf )E(Zc , zf )g = zf E(Zc , zf ) =,...,zf 1zf P M

T=2

N

Kn Gen ,

n=1

2 ( % H = 2zf E(Zc , zf ) = zf g = 2

N

Kn zf G en + Kn GGT KTn ,

:LHO;

n=1

en '%

n=9

"(T[1][L] " en = en , . . . , en @ Kn M P M PKn = (cn,1 , . . . , cn,1 , cn,2 , . . . , cn,2 , . . . , cn,P , . . . , cn,P , cn,1 , cn,2 , . . . , cn,P );%

$ %

$%

$M1 veces

M1 veces

M1 veces

' G M P L

[1] [1]

x1 p1

[1][1]xM1 p1

[1] [1] x p1 PG=

[1][1]x M1 pP p[1]P

[1]

pP

[2] [2]

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[2]

xM1 p1

[2] [2]

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[2]

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[2]

pP

..................

[L] [L]

x1 p1

[L][L] xM1 p1 [L] [L] x1 pP , [L][L] xM1 pP [L]pP

[L]

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DEP Salida (dBm/Hz)

DEP Entrada (dBm/Hz)

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Amplitud (V)

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Informacin Mutua Cuadrtica

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0.080.060.040.0200

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Respuesta al Impulso

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sR [n]

d3 [n]

-

s[n]-

T

BChT (t)s[n]

-

d+3 [n]

-

%T

hR (t)

-

1

tT

sT (t)

-

1

""s" ? s -

tT

sR [n]

d3 [n]

-

s[n]-

T

BC

. )) ; 2 =" ; 2

! # !

'

kd+3 [n] = (1 z1 )z1 u[n],kdu[n].3 [n] = (1 z1 )z1

2 d+3 [n] = d3 [n]" D3+ () = D3 () '" " ! " '

%" ?= %

) " ! =# ! 2 7 MM 8 ! 2 %" hT (t) ' hR (t)"

! -, :1 % 1 ;

tn ,hT (t) =TnZ

t1hR (t) = hT (t) =d3 [n] + n .T

1d+3 [n]

nZ

2 #

! -, sinc2 (T /2),1 z1 ejTs[n]"

HT () = TF {hT (t)} = (1 z1 )

' 2 "% S2 "

" sT (t)" %

323 22ST (f ) = S |HT (f )| = STT

sinc4 (f T )2 + cos(2f T )

.

HJ4

M 9

NRZNo adaptado, m=3Adaptado, m=3Adaptado, m=5

0

DEP (dB)

1020304050600

1

23Frecuencia normalizada fT

4

5

. )+ 2

# +,S

) 7 MJ ) sT (t) m = 3 : ; '1 # % ' # +,S % 2

# 3= J ? 1 +,S" 2" # 2 "

" 20 ? 1 +,S "

# (f T )4" 2 +,S

(f T )2) % 2" % :#

' ;" ! '

:ML;" ' # : 7ML; m = 5"

H3 (f ) =

3 sinc4 (f T )2= |HT (f )| .2 + cos(2f T )

7 " 7 MO" 1 # ' # 0,5 8 7 MO 1 ! # L

M6

2

2

1.5

1.5

1

1

0.5

0.5

0

0

0.5

0.5

1

1

1.5

1.5

21

0.5

0

0.5

1

21

21.5

1

1

0.5

0.5

0

0

0.5

0.5

1

1

1.5

1.5

0

E

0

2

0.5

0.5

H

1.5

21

HJI

0.5

r = 0,5

1

21

0.5

0.5

1

0.5

1

I

0

E

r = 0,25

. ) 1" " ' 2 ! 7, 1 2 H(z) = # M = 2H(z) =

1 1 + 8z 1 + 23z 2 + 32z 3 + 23z 4 + 8z 5 + z 6 ;48

" ! # '

P = 7 $

" 1 = ! % ! 8? P KOQ

7 = " 2 # 1

# 2

" =

L = 5 F (z) = 16 1 8z 1 + 20z 2 8z 3 + z 4 8 11 21z 2 + 164z 4 + 288z 5 + 164z 6 21z 8 + z 10 " 2 G(z) = 288 ! +'2:4; # % ) 0 = 2" 1 " !

L = 5:2

2 ! # '+'2:4;;" " !2" " ! g[n] +'2:4; # ' %7 " ! ' #

G(z) = R(z)R(1/z)

HK6

M 9 10

Respuesta en frecuencia (dB)

010203040506070800

PSDBPSQRC (r=0.25)0.1

0.20.30.4Frecuencia normalizada fT

0.5

. )#+ , # ! # =?

! ?= # ! -,

M = 2# PO4Q 2 1

g[n] ' 7 % ln (G(2k/N )) #

'

77 W

N = 214 " !

R(z) = 0,4623 + 0,8369z 1 + 0,2584z 2 0,1373z 3 0,0136z 4 + 0,0075z 5" 2 ! # R(1/z) ! %" 0 !

:-, 1

; L !" #

M = 2 ' 3 0 r = 0,25 ) !

R (z) = 0,0454 + 0,4397z 1 + 0,7554z 2 + 0,4397z 3 0,0454z 4 ' # 7 MHJ 1

! R(z)$ ! 9

!

9 % " ! P055" 54Q

P =

n=0

|g[nM ]|

|g[0]|

,

g[n] 1 ! ' ) 1 " g[n] %

# #

% D E

MO

HKL

r[n] % C1 r[n] 8 P = 9,6 109" % 3

3

9

# "

! -, 5 ! P = 0,2637" % 2

# ' %

)+*$ F"

1 2

! # -, ) "

! % 1" 0 ! -, " 1 " ! -, P = 19 !" 3 0

r = 0,35 ' # M = 3) = #

! -,

"

1 ! L = 9 8

%

: %;" 2"

! 0 %

" 1 ! % L = 31 !" 2 # " R(z)" 4L !

1" 9

" ! -, 4L ! 8 ! ' #= ! :-, " ?= ' -, 4L !; 7 MHO % 2 % ! #

0 0 0

! -, % 2

0 % " ?= P = 2,1 107" 2 -,

P = 0,0341

#1 ,

) 0 00 0 3 ' #

' 2 ' =

0 9 3 $ E

2 ! % H(z) H

9:"!BM C !

L=2

L+P 22M

+1

HKM

M 9 10

0.7

Modulo respuesta frecuencial (dB)

0.6Respuesta al impulso

0.5

20

0.4

30

0.3

40

0.2

50

0.1

60

0

70

0.10.2

0

10

5

10

n

15

20

25

800

0.1

0.20.30.4Frecuencia normalizada fT

0.5

. )# P85H" 85HQ; 2 0 =0 :$G? I% 9

; PGKO" R 54" * 56Q

% %

"

' " 1 '" "

#(

7 8

# 5 &

$

# = # 2 # :

H"

, 7 ? , 0

G8 L =; 2 3

" !

45M

J " ' #

$ 2 0k>0

22p+1 22(p+1) 1Bp+1 ,=(2p + 2)!

% '

n=9 ?

" Bn " % PJ5" H5K ' HHLQ

'& ( ) 9 3 % '

= % " % " 3 ! =

, ) +

3 :64O;

! =

d [n] =

(1 z1 ) |n|z ,(z1 + 1) 1

" 2 [n]" [n] = d [n] d [n] = d [n] d [n],

%

|n|

[n] =

(1 z1 )z1 (|n| + 1 + (1 |n|)z12 ).(z1 + 1)3

:H;

" 9 #

! = = d(2) "(z1 1)3 |n|1z, n = 0;(z1 + 1) 1d(2)[n]=d[n]([n+1]2[n]+[n1])=2 2(z1 1) ,n = 0;(z1 + 1)

4HH

4H4

9 < ' % =

' " (2) " 3:4;

(2)(2)(2)(2)(2) [n] = d [n] d [n] = d [n] d [n],

%

(2) [0] =(2)(2) [1] = [1] =

2(d(2) [n]) =

n=

2(z1 + 3)(z1 1)4,(z1 + 1)3

(2)d(2) [n]d [1 n] =

n=

:4;

2(z1 + 2)(z1 1)5.(z1 + 1)3

:4;

, 0

+ +

N M

" x[n]" # 0 = 2/T0 = 2M/N

Mx[n] = A sin (0 n + 0 ) = A sin 2 n + 0 ,N

0 M N 1 ' 0 0 28 = s (t) ' :2 2% %" ;

" :64I;" < (s (t)) = M1T0

1(s (t)) dt =NMT02

NN

2c[n] (t n)

dt.

n=1

2 % [n, n + 1] % (t n + 1)" (t n)" (t n 1) ' (t n 2)" < (s (t)) = N1

N

n=1

n+1

(c[n 1] (t n + 1) + c[n] (t n)

n2

+c[n + 1] (t n 1) + c[n + 2] (t n 2)) dt.

" " c[n] = c[n + kN ]"

4 <

4HI

< (s (t)) = N1+

2N

N

c2 [n]

n=1N

2

( (t)) dt

c[n]c[n + 1]

n=1

(t) (t 1) dt

N2 +c[n]c[n + 2] (t) (t 2) dtN n=1 N2 +c[n]c[n + 3] (t) (t 3) dt.N n=1

:I;

" ! =

c[n] = f [n] D (z)|z=ej0 = A sin (0 n + 0 )

3,3 + (cos 0 1)

2 :I; %

N

N

9A2c[n]c[n + k] =sin (0 n + 0 ) sin (0 (n + k) + 0 )[3 + (cos 0 1)]2 n=1n=1

=

9NA2cos (0 k).2 [3 + (cos 0 1)]2

7 " % = N =

M

A ' #0 2

< (s (t)) = A2

9 (I0 + 2 cos (0 )I1 + 2 cos (20 )I2 + 2 cos (30 )I3 ),[3 + (cos 0 1)]2

:6;

In =

(t) (t n) dt,

:L;

4H6

9 < ' % =

%

I0 =I1 =I2 =I3 =

313 702 +140,210357 +2386 6725 +9804 2803 +842 14+1,6304623 +1052 +105,6304007 22966 +47045 30804 +19603 5882 +987,504043131260 ,977 +4486 6725 +5604 2803 +842 14+1,12604

0,

(21)750404 ,

0 12 ;12 1;0 12 ;12 1;0 12 ;12 1;0 12 ;12 1;

:6;

% =

# #" "'

%" " 1 + 2 cos (2)I2 + 2 cos (3)I3 )< (, ) = 9 (I0 + 2 cos ()I[3 + (cos 1)]2

' % :L;)% 3" 2 % =

= # %

= 2 % # " 3 2 :0,7 < < 1 ' > 0,9 7 6H4;"

% ) " #

#" % = # ) " %

% =

: "

;" 7 " X[k]" " x[n]"

3< (s (t)) = N12

N1

2

|X[k]|

< (, = 2k/N ) ,

k=1

2 ' 9 :k = 0 7 ; ' % " ' 2 = 3 0 " %

I <

4HL

, 0

+ +

% =" < (S )"

# 6MI" % =

RS ( ) =

RC [n]

n=

(t n + ) (t) dt.

8 % =

3 = 0

< (S ) =

RC [n]

n=

(t n) (t) dt.

# =" 2 %" ' % % :L;"

|n| > 3

< (S ) = RC [0]I0 + (RC [1] + RC [1]) I1+ (RC [2] + RC [2]) I2 + (RC [3] + RC [3]) I3 ,

'" 2 RC [n] 9 : ;" 3:M;< (S ) = RC [0]I0 + 2RC [1]I1 + 2RC [2]I2 + 2RC [3]I3 , % % = %

:L; '

" RC [n]

RC [n] = RX [n] [n],:J;

% [n] % :H;) #" %

#

" SX ()"

' # = :6IJ;1< (S ) = 2=

12

+

+

1SS ()d =2SX ()

+

2

SX () |H ()| d

9 sinc4 (/2) sinc4 (/2)[3 + (cos 1)]2

d,

2 % '

4HM

9 < ' % =

8' &

*

2

+

X[n] "

! = RC [n] = X2 [n]" 2 %

" 0 :M; ' :H;" < (S ) = X2 (z(1 +z1)1 )31

/0(1 + z12 )I0 + 4z1 I1 + 2z12 (3 z12 )I2 + 4z13 (2 z12 )I3 .

3 0 12 " 5

4

3

2

1 + 1079z1 + 595z1 + 70z1 35< (S ) = 2X2 35z1 + 488z105(z. :O; 1)2 (z + 1)31

12

1"

1

/

2

< (S ) = 1632960z 4(zX1)2 (z1

1

31 + 1)

z116 + 71z115 + 2175z114

+ 37387z113 + 391727z112 + 2530731z111 + 9591043z110+ 19088975z19 + 21519117z18 + 15050501z17 + 4966381z16

01908879z15 1772155z14 68663z13 4143z12 139z1 2 .

:O;) 7 H %

% = ?

0 !" 3 " 2d< (S ) 0, z [2 + 3, 0], 2

dz1

d< (S ) 0,d

1

[0, 1].

) " % =

# +

8' &

*

# 0" =

2RX [n] = Xcos (0 n),

n Z, 0 0 ,

6 & 0.9

0.4

d Var(S ) / d

0.3

0.8

Var(S )

0.85

0.75

0.2

0.1

0.70.650

4HJ

0.2

0.4 0.6

0.8

1

00

0.2

0.4 0.6

0.8

1

. # < = :2=

; ' % :0;

'" :J; ' :6IM;" ! = 2

2RC [n] = Xcos(0 n) |Hd ()| ,

2 % " :M;" %

2 cos (20 )I2 + 2 cos (30 )I3 )< (S ) = X2 9 (I0 + 2 cos (0[3)I1++(cos. 1)]20

%" 3 2% %

" 2

9" " % # 2 % ' #" " " % 7 6H4"

%

,! 2

+

+% " = N = M " x[n]" # 0 = 2/T0 = 2M/N

Nx[n] = A sin (0 n + 0 ) = A sin 2 n + 0 ,M

0 M N 1 ' 0 0 2

4HO

9 < ' % =

-

x(t)

-

%T

x[n]

7

d [n]

-

%T

-

7

s (t)

(t)

^

? J(t)

s

-

%T

-

8

=6

-

L(t)

(t)

. $ )2

=

# ! J(t) :%7 4; # =" s (t)" '

" L(t)" "J(t) = s (t) L(t),

# 2 =

J(t)" 2 " "r (s (t)) = M1T

0

[s (t) L(t)]2 dt

MT0

= < (s (t)) + < (L(t))

2M T0

s (t)L(t)dt.

:K;

MT0

) 3" 9 %

:6; , 9 "

:64I; ' 00 2 2 %

# (t) = (t) |=0 # "

1s (t)L(t)dtM T0 MT0 N

N

1=c[n] (t n)x[n] (t n) dt.N N n=1n=1

IA =

(

# (t) ' (t)" '

x[n] ! = c[n]"2 !

= 0C E B'4C ; (L(t)) = A2 (2 + cos 0 )/6

B

7

6 &

4HK

N2

1IA =c[k] f [k + j] (x k) (x k + j) dxNj=2k=1

NNNIA0 2IA1 2IA2 =c[k]f [k] +c[k]f [k + 1] +c[k]f [k + 2],NNNk=1

k=1

IAn =

' %

:H5;

k=1

(x) (x n) dx = IAn ,

:HH;

33 102 + 40,6023 + 52 + 10,=603.=120

IA0 =IA1IA2

8 ! = % c[n] = x[n] D (z)|z=ej0 = A sin (0 n + 0 )

2 :H5; %

N

c[n]x[n + k] =

n=1

=

3,3 + (cos 0 1)

N

3A2sin (0 n + 0 ) sin (0 (n + k) + 0 )3 + (cos 0 1) n=1

A2 N3 cos (0 k).2 3 + (cos 0 1)

7 " :H5; IA =

3A2(IA0 + 2 cos (0 )IA1 + 2 cos (20 )IA2 ) .2 3 + (cos 0 1)

:H4;

" ' :H4; ' :6; :K;"

=

A ' # 0 %

22 cos (20 )I2 + 2 cos (30 )I3 )r (s (t)) = A2 9 (I0 + 2 cos (0[3)I1++(cos 1)]20

2

A 2 + cos 023A2 6 (IA0 + 2 cos (0 )IA1 + 2 cos (20 )IA2 ),23 + (cos 0 1)+

:HI;

445

9 < ' % =

In IAn %

:L; ' :HH;" % ) !%" = =

" 3% " # '

%

% " 31 + 2 cos (2)I2 + 2 cos (3)I3 )r (, ) = 9 (I0 + 2 cos ()I[3 + (cos 1)]2

6 (IA0 + 2 cos ()IA1 + 2 cos (2)IA2 ) 2 + cos 0+,3 + (cos 1)3

:H6;

2 % ' %

3 # =

% 2 #" "

[ 3 2 3 #

+'2 ' % 3 :% 7 6H4;

" 2 # " x[n]"

=

7 " X[k]" ' # :H6;="

r (s (t)) = N12

N1

2

|X[k]|

r (, = 2k/N ) ,

k=1

," 2

+

) = " RX [n] '

|SX ()|2 "

J(t) 7 4 0 !

2 2

=

X[n]"

hJ (t) = (t) (t) ,

' #2

HJ () = H () H () = sinc (/2)

3 sinc 2 (/2)1 .3 (1 cos )

"

" RJ ( )" RJ ( ) = RX [n] (hJ ( ) hJ ( )) ,

L &

44H

'" :L; ' # (x) ' (x) 2 %=" RJ ( ) = RX [n] [ ( ) ( ) + ( ) ( ) 2 ( ) ( )] . :HL;7 " 3 = 0"

=

+

#* 2

' " " RX [n] =" 0 :HL;"

= 2X[n]"

r (S ) = X2 RJ (0) = < (S ) + < (S )|=0 2X2

( ) ( ) d,

< (X ) % = ' < (X )|=0 % @ ' % % :O; ) " 9

(t) (t) dt = 2=2

1

0

nZ

d [n] (t n) (1 n)dx

d [n]

nZ

1

0

(1 t) (t n) dx,

"

# (t) " % n = 1, 0, 1, 2

= #

z1

! =

0 12 " 6

5

4

3

2

1 + 380z1 + 1985z1 + 895z1 + 140.r (S ) = 2X2 z12 35z1 35z1 376z105(z 1)4 (z + 1)31

12

1"

1

/

2

r (S ) = 1632960z 4(zX1)4 (z

z118 + 69z117 + 2034z116+11+ 33108z115 + 319128z114 + 1784664z113 + 3832668z112+ 2437620z111 8374158z110 25010630z19 31920144z18 23353956z17 10406256z16 2787792z15 550332z14060516z13 3867z12 135z1 2 ;1

1)3

444

9 < ' % =

0.035

0.06

0.03

0.05d Ar(S ) / d

Ar(S )

0.0250.020.0150.01

0.030.020.01

0.00500

0.04

0.2

0.4 0.6

0.8

1

00

0.2

0.4 0.6

0.8

1

. & & = :=2; ' % :0;

# 2" ' % 7 I" = z1 " 2 " " " = # % +

#*

%

% ' % = =

" # 0 ' % X2

3 :HI; ) 2 cos (20 )I2 + 2 cos (30 )I3 )r (S ) = X2 9 (I0 + 2 cos (0[3)I1++(cos 1)]20

2 + cos 0+36(I+ 2 cos (0 )IA1 + 2 cos (20 )IA2 )A0 F2.3 + (cos 0 1)F2

)* "7 % + '8 3 = # " (t)"

% [t0 , t1 , t2 , t3 , t4 ] ' % = [h1 , h2 , h3 , h4 ]" % :6MO;

% :6J5;) 2 0 0,5

t1 h1 < t < t1 "

(t) =

t = t1 "

(t) =

(t t1 + h1 )3.2 h1 (h1 + h2 )(h1 + (3 2)h2 + h3 )

h21.(h1 + h2 )(h1 + (3 2)h2 + h3 )

t1 < t < t1 + h2 "

(t) =

3 h21 h2 + 32 h1 h2 (t t1 ) + 3h2 (t t1 )2 (t t1 )3.2 h2 (h1 + h2 )(h1 + (3 2)h2 + h3 )

44I

446

t = t1 + h2 "

(t) =

h4 + 2h3 + 3(t + t3 h3 )(h4 + (3 2)h3 + h2 )[h4 + (3 )h3 + (3 )h2 + h1 ](t + t2 + h3 )3. 2 h3 (h3 + h2 )(h4 + (3 2)h3 + h2 )(h3 + (3 2)h2 + h1 )

t = t2 + h3 "

(t) =

h1 + (3 )h2(h1 + (3 2)h2 + h3 )h1 + (3 )h2 + (3 )h3 + h4h2 .(h2 + h3 )(h1 + (3 2)h2 + h3 )(h2 + (3 2)h3 + h4 ) 2

t2 < t < t2 + h3 "

(t) =

h1 + 2h2 + 3(t t1 h2 )(h1 + (3 2)h2 + h3 )[h1 + (3 )h2 + (3 )h3 + h4 ](t t2 + h2 )3. 2 h2 (h2 + h3 )(h1 + (3 2)h2 + h3 )(h2 + (3 2)h3 + h4 )

t = t2 "

(t) =

h1 + (3 4)h2.(h1 + (3 2)h2 + h3 )

t2 h2 < t < t2 "

(t) =

h1 + 2h2 + 3(t t1 h2 ).(h1 + (3 2)h2 + h3 )

t = t2 h2 "

(t) =

h1 + 2h2.(h1 + (3 2)h2 + h3 )

t1 + h2 < t < t2 h2 "

(t) =

9 )3 = #

h4 + (3 4)h3.(h4 + (3 2)h3 + h2 )

t2 + h3 < t < t3 h3 "

(t) =

h4 + 2h3 + 3(t + t3 h3 ).(h4 + (3 2)h3 + h2 )

H )3 = #

t = t3 h3 "

(t) =

t3 h3 < t < t3 "

(t) =

h24.(h4 + h3 )(h4 + (3 2)h3 + h2 )

t3 < t < t3 + h4 "

(t) =

3 h24 h3 + 32 h4 h3 (t + t3 ) + 3(t + t3 )2 (t + t3 )3.2 h3 (h4 + h3 )(h4 + (3 2)h3 + h2 )

t = t3 "

(t) =

h4 + 2h3.(h4 + (3 2)h3 + h2 )

(t + t3 + h4 )3.2 h4 (h4 + h3 )(h4 + (3 2)h3 + h2 )

t t1 h1 t t3 + h4 "

(t) =0.

0,5 1

t1 h1 < t < t1 "

(t) =

t = t1 "

(t) =

(t t1 + h1 )3.2 h1 (h1 + h2 )(h1 + (3 2)h2 + h3 )

h21.(h1 + h2 )(h1 + (3 2)h2 + h3 )

t1 < t < t2 h2 "

(t) =

3 h21 h2 + 32 h1 h2 (t t1 ) + 3h2 (t t1 )2 (t t1 )3.2 h2 (h1 + h2 )(h1 + (3 2)h2 + h3 )

44L

44M

t = t2 h2 "

(t) =

h1 + 2h2 + 3(t t1 h2 )(h1 + (3 2)h2 + h3 )[h1 + (3 )h2 + (3 )h3 + h4 ](t t2 + h2 )3. 2 h2 (h2 + h3 )(h1 + (3 2)h2 + h3 )(h2 + (3 2)h3 + h4 )

t = t2 "

(t) =

h1 + 2h2(h1 + (3 2)h2 + h3 )[h1 + (3 )h2 + (3 )h3 + h4 ](1 2)3 h22.+ 2 (h2 + h3 )(h1 + (3 2)h2 + h3 )(h2 + (3 2)h3 + h4 )

t1 + h2 < t < t2 "

(t) =

3 h21 h2 + (1 )h22 (32 h1 + (1 )(4 1)h2 )2 h2 (h1 + h2 )(h1 + (3 2)h2 + h3 )3h2 (2 h1 + (1 )(3 1)h2 )(t t2 + h2 )+2 h2 (h1 + h2 )(h1 + (3 2)h2 + h3 )3h2 (2 1)(t t2 + h2 )2 (t t2 + h2 )3+2 h2 (h1 + h2 )(h1 + (3 2)h2 + h3 )[h1 + (3 )h2 + (3 )h3 + h4 ](t t2 + h2 )3. 2 h2 (h2 + h3 )(h1 + (3 2)h2 + h3 )(h2 + (3 2)h3 + h4 )

t = t1 + h2 "

(t) =

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t2 h2 < t < t1 + h2 "

(t) =

9 )3 = #

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t2 < t < t3 h3 "

(t) =

h4 + 2h3 + 3(t + t3 h3 )(h4 + (3 2)h3 + h2 )[h4 + (3 )h3 + (3 )h2 + h1 ](t + t2 + h3 )3. 2 h3 (h3 + h2 )(h4 + (3 2)h3 + h2 )(h3 + (3 2)h2 + h1 )

H )3 = #

t = t3 h3 "

(t) =

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t3 < t < t3 + h4 "

(t) =

3 h24 h3 + 32 h4 h3 (t + t3 ) + 3(t + t3 )2 (t + t3 )3.2 h3 (h4 + h3 )(h4 + (3 2)h3 + h2 )

t = t3 "

(t) =

3 h24 + (1 )h3 (32 h4 + (1 )(4 1)h3 ).2 (h4 + h3 )(h4 + (3 2)h3 + h2 )

t2 + h3 < t < t3 "

(t) =

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t = t2 + h3 "

(t) =

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t3 h3 < t < t2 + h3 "

(t) =

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(t + t3 + h4 )3.4 (h4 + h3 )(h4 + (3 2)h3 + h2 )

t t1 h1 t t3 + h4 "

(t) =0.

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