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5 ' H 64 ? =" (t)" %
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MOJHJIJJJOOIO6OMK5K6H5HH54H5IH5LH5KH5KHH5HHHHHIHHL
LH 7 % = H44
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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 % : ;
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% =
:' ' % 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
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[2]
xM1 p1
[2] [2]
x1 pP[2]
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[L] [L]
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50607080901003
DEP Salida (dBm/Hz)
DEP Entrada (dBm/Hz)
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1
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2
3
2
1
01Frecuencia (MHz)
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L
=
25
Entrada GananciaSalida
Amplitud (V)
20
15
10
5
0250
260
270280tiempo (s)
290
300
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L
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0.16
Informacin Mutua Cuadrtica
0.14
BW =2 MHziBW =6 MHziBW =9 MHzi
0.120.1
0.080.060.040.0200
1
2
3
k
4
5
6
7
. *#& # y [n] ' x [n k] #I
I
k *7=-R
# 0
Pi = 9
? '
%% 1 '
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(Ids ) (Vgso , Vdso )"
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1 L = 1000
1
8
%
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1
1 %
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ds
dIdsdVds
dIdsdVgs
25,5030,6231,4130,3227,1732,4433,1432,0228,0933,0433,7132,6628,8432,92
14,009,5218,9415,7612,6210,4418,6918,2013,7111,5118,8717,4410,090,8674
14,4013,0416,2816,4114,7512,9617,6717,5314,6614,3417,4218,1913,7916,53
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() 0
54(:4 30 ;
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=
0.060.05
MedidasModelo PWL
vgs=0.50
vgs=0.25
0.03
vgs=0.00
Ids
0.04
0.02
vgs=0.25
0.01
vgs=0.50
00.010
1
V2
3
4
ds
0.06
0.08
0.05
0.07
0.05
0.03vgs=0.25
0.02vgs=0.25
0.01
0.010.020
0.04
ds
vgs=0.00
0.010
vgs=0.00
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vgs=0.50vgs=0.25
I /V
0.02
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0.040.03
vgs=0.50vgs=0.25
0.06
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01
V
2
3
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4
0.010
1
V
2
3
4
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HJ5
M 9 1m=3m=5m=9m=
0.60.40.200.20.45
m=3m=5m=9m=
0Respuesta en Frecuencia (dB)
Respuesta al Impulso
0.8
20406080100120140160
0Tiempo normalizado t/T
5
1800
1
23Frecuencia normalizada fT
4
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-
1
""s" ? s -
tT
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) =
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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 t1 h1 t t3 + h4 "
(t) =0.
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