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It is a capital mistake to theorize before you have all the evidence.
It biases the judgment.
– SHERLOCK HOLMES (
, 1887)
m × mg
MCW
j
2 × 8 (m = 2, g = 4)
2 × 8
4 × 16
2 × 128
2 × 8
2 × 8 2 × 128 f
D
T
s
= 0, 002
2 × 8 2 × 128 f
D
T
s
= 0, 002
2×128
2 × 8 2 × 128
f
D
T
s
= 0, 002
2 × 8 2 × 128
f
D
T
s
= 0, 004
2×8
2 × 128 f
D
T
s
= 0, 002
2×8
2 × 128 f
D
T
s
= 0, 004
2×8 4×16 f
D
T
s
= 0, 002
2×8 4×16 f
D
T
s
= 0, 002
2×8
f
D
T
s
= 0, 002
4 ×16
f
D
T
s
= 0, 002
2 × 8 4 × 16
f
D
T
s
= 0, 002
{2, 4, 6}
2 ×8
2 × 8
2 × 128
2 × 128
2 × 8
2 × 128
y
x
8
2 × 128
2 × 128
y
0
x
8
A = (a
s
k
)
a
0
a
s
, s ≥ 0
a
s
k
C
MCW
E[·]
E
b
E
b
/ℵ
0
E
C
E(I
d
, L) I
d
E
s
e
n
D
f
D
T
s
f
0
G
ν
i
(z)
GF (q) q
g
I(X; Y )
I
d
J
0
(·)
K
L · I
d
L
2
(R)
m
n
[nT
s
, (n + 1)T
s
)
n(t)
P
erro
P (˜y
n
| y
n
) ˜y
n
y
n
p(t)
p(y
n
) y
n
Q(·)
R
b
R(∆f)
R(∆t)
R
s
r(t)
S( )
S(τ)
s(t)
T
m
T
0
W[A] A
x
x
n
y
n
y
n
y
j
n
z
i
i
α(t)
δ
x,y
µ
ν
i
σ
2
τ
ϕ
jk
(x)
ψ
s
jk
(x)
ℵ
0
AP K
ASK
AW GN
BER
BP SK
CSI
DEP
EMQ
ISI
i.i.d.
MAP
MCW
P EI
P SK
RI
SNR
ST BC
V LSI
v.a.
W SS US
K
A = (a
s
k
) m ≥ 2 mg
A =
a
0
0
, . . . , a
0
mg−1
a
1
0
, . . . , a
1
mg−1
a
m−1
0
, . . . , a
m−1
mg−1
,
A m g
mg−1
k=0
a
s
k
= mδ
s,0
, 0 ≤ s ≤ m − 1
mg−1
k=0
a
s
′
[k+mr
′
]
a
s
[k+mr]
= mδ
s
′
,s
δ
r
′
,r
, 0 ≤ s
′
, s ≤ m − 1,
0 ≤ r
′
, r ≤ g − 1
[k+mr] k+mr mg ¯a
a δ
x,y
δ
x,y
=
1
x = y
0
m
√
m
m
m
m
±1
mg−1
k=0
a
s
k
= m
√
gδ
s,0
,
mg−1
k=0
a
s
′
[k+mr
′
]
a
s
[k+mr]
= mgδ
s
′
,s
δ
r
′
,r
.
±1 2 × 2
1 1
1 −1
m = 2
1 1 1 −1 1 1 −1 1
1 1 1 −1 −1 −1 1 −1
.
m = 2
1 1 1 1 1 −1 1 −1 1 1 1 1 −1 1 −1 1
1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 1 −1 1 −1
1 1 −1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 −1
1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 −1 1
.
x
n
∈ {+1, −1}
m g
x
n
S/P
Conv.
Fonte
m−1
MCW
MCW
j
0
MCW
x
pm
x
pm+j
x
(p+1)m−1
j
pm+q
y
0
pm+q
y
m−1
pm+q
y
D
−j
m
0
D
m
m
−(m−1)
D
a
j
m
a
j
2m
a
j
(g−1)m0
a
j
j
a
gm−1
j
a
3m−1
j
a
2m−1
j
a
m−1
j
a
m+q
a
j
(g−1)m+q
a
j
q
a
j
2m+q
x
pm+j
D
−1
D
−1
D
−1
D
−1
D
−1
D
−1
D
D
D
−1
−1
−1
pm
j
pm+q
j
(p+1)m−1
j
pm+q
j
y
y
y
y
MCW
j
q m − 1
m × mg
MCW
j
x
n
m
X
pm+j
:= {x
pm+j
}
p∈Z
, 0 ≤ j < m
j X
pm+j
MCW
j
n = pm+q
p ∈ N = {0, 1, 2, 3, 4, . . .} q ∈ {0, 1, . . . , m − 1} j MCW
j
y
j
pm+q
m
MCW
j
MCW
j
m REG
q
g mg j
m g q
REG
q
j
n = pm + q m y
j
pm+q
, 0 ≤ j ≤ m − 1
q m MCW
j
y
j
pm+q
n = pm + q q
MCW
j
y
j
pm+q
=
g−1
l=0
a
j
lm+q
x
(p−l)m+j
.
mg
mg
K
K := mg
m
n = pm + q
y
pm+q
=
m−1
j=0
g−1
l=0
a
j
lm+q
x
(p−l)m+j
.
2×8 y
j
n
y
n
A =
a
0
0
a
0
1
a
0
2
a
0
3
a
0
4
a
0
5
a
0
6
a
0
7
a
1
0
a
1
1
a
1
2
a
1
3
a
1
4
a
1
5
a
1
6
a
1
7
.
D
−1
D
−1
D
−1
D
−1
D
−1
a
1
1
a
3
1
a
5
1
a
1
4
a
1
2
a
1
0
D
−1
D
−1
D
−1
D
−1
−1
D
D
−1
D
−1
a
0
1
a
0
3
a
0
5
a
0
4
a
0
2
0
a
0
a
0
6
2p+1
0
y
1
2p
y
2
2
x
2p
x
n
2p
y
2p+1
y
2p
y
D
−1
a
1
a
1
6
a
0
7
7
S / P
Conv
D
0
0
1
2p+1
y
n
Y
x
2p+1
2 × 8 (m = 2, g = 4)
2 × 8
nT
s
y
0
n
y
1
n
y
n
= y
0
n
+ y
1
n
a
0
0
x
0
a
1
0
x
1
a
0
0
x
0
+ a
1
0
x
1
a
0
1
x
0
a
1
1
x
1
a
0
1
x
0
+ a
1
1
x
1
a
0
2
x
0
+ a
0
0
x
2
a
1
2
x
1
+ a
1
0
x
3
a
0
2
x
0
+ a
0
0
x
2
+ a
1
2
x
1
+ a
1
0
x
3
a
0
3
x
0
+ a
0
1
x
2
a
1
3
x
1
+ a
1
1
x
3
a
0
3
x
0
+ a
0
1
x
2
+ a
1
3
x
1
+ a
1
1
x
3
a
0
4
x
0
+ a
0
2
x
2
+ a
0
0
x
4
a
1
4
x
1
+ a
1
2
x
3
+ a
1
0
x
5
a
0
4
x
0
+ a
0
2
x
2
+ a
0
0
x
4
+ a
1
4
x
1
+ a
1
2
x
3
+ a
1
0
x
5
a
0
5
x
0
+ a
0
3
x
2
+ a
0
1
x
4
a
1
5
x
1
+ a
1
3
x
3
+ a
1
1
x
5
a
0
5
x
0
+ a
0
3
x
2
+ a
0
1
x
4
+ a
1
5
x
1
+ a
1
3
x
3
+ a
1
1
x
5
a
0
6
x
0
+ a
0
4
x
2
+ a
0
2
x
4
+ a
0
0
x
6
a
1
6
x
1
+ a
1
4
x
3
+ a
1
2
x
5
+ a
1
0
x
7
a
0
6
x
0
+ a
0
4
x
2
+ a
0
2
x
4
+ a
0
0
x
6
+ a
1
6
x
1
+ a
1
4
x
3
+ a
1
2
x
5
+ a
1
0
x
7
a
0
7
x
0
+ a
0
5
x
2
+ a
0
3
x
4
+ a
0
1
x
6
a
1
7
x
1
+ a
1
5
x
3
+ a
1
3
x
5
+ a
1
1
x
7
a
0
7
x
0
+ a
0
5
x
2
+ a
0
3
x
4
+ a
0
1
x
6
+ a
1
7
x
1
+ a
1
5
x
3
+ a
1
3
x
5
+ a
1
1
x
7
y
n
y = x · C
MCW
y C
MCW
m
C
MCW
C
MCW
C
MCW
C
MCW
=
a
0
0
a
0
1
a
0
2
··· a
0
7
a
1
0
a
1
1
a
1
2
··· a
1
7
a
0
0
a
0
1
a
0
2
··· a
0
7
a
1
0
a
1
1
a
1
2
··· a
1
7
y
n
mg
mg
y
n
m mg
m
y z
j
, j ∈ {0, 1, . . . , m−
1}
a
j
i = m(g + p) − 1 p ∈ N
z
j
i
=
mg−1
k=0
a
j
(mg−1)−k
y
i−k
=
mg−1
k=0
m−1
j
′
=0
g−1
l=0
a
j
k
a
j
′
k−lm
x
j
′
+lm+i−(mg−1)
z
j
i
= x
j+i−(mg−1)
mg−1
k=0
a
j
k
a
j
k
= mgx
j+i−(mg−1)
x
j+i−(mg−1)
−1 z
j
i
= −mg +1
z
j
i
= +mg
ˆx
j+i−(mg−1)
= (z
j
i
)
y
n
m
g
y
n
∈ {−mg, −mg + 2, . . . , −mg + 2k, . . . , −2, 0, 2, . . . , mg − 2, mg}
mg + 1
Pr(y
n
= 2k − mg) =
mg
k
0, 5
mg
, 0 ≤ k ≤ mg.
mg
y
j
n
g + 1
y
j
n
∈ {−g, −g + 2, . . . , −g + 2k, . . . , −2, 0, 2, . . . , g − 2, g}
y
j
n
Pr(y
j
n
= 2k − g) =
g
k
0, 5
g
, 0 ≤ k ≤ g
g
y
n
m
m = 2
1/g
Codificador
Wavelet
Decodificador
Wavelet
Receptor
Fonte
Modulador
Demodulador
Antena
Canal
s
n
x
n
ˆx
n
y
n
ˆy
n
r
n
x
n
C
MCW
y
n
mg + 1
y
n
mg+1 nT
s
y
n
s(t) y
n
mgT
s
r
n
(t)
r
n
(t) = s
n
(t) + n
n
(t), nT
s
≤ t ≤ (n + 1)T
s
,
n
n
(t)
ℵ
0
/2
r
n
(t) r
n
= y
n
+n
n
y
n
n
n
ℵ
0
/2
ℵ
0
= mg · 10
−0,1(
E
b
ℵ
0
)
dB
E
s
/ℵ
0
E
b
/ℵ
0
r
n
2 ×8 4 ×16
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
0 2 4 6 8 10
Prob. de Erro de Bit
Eb/No (dB)
BPSK
PAM-MCW 2x8
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
0 2 4 6 8 10
Prob. de Erro de Bit
Eb/No (dB)
BPSK
PAM-MCW 4x16
4 × 16
y
n
2.14
Desentrelacador
LMS
Estimador
.
.
.
Codificador
Wavelet
Entrelacador
Modulador
Receptor
Decodificador
Wavelet
.
Demodulador
Canal
Antena
Fonte
s
n
ˆ
s
n
x
n
ˆx
n
y
n
¯y
n
ˆ
¯y
n
ˆy
n
r
n
r
n
ˆα
n
E(I
d
, L)
I
d
L
L · I
d
y
y
n
I
d
= 4
y
0
y
1
y
2
y
3
y
4
y
5
y
6
y
7
y
8
y
9
y
10
y
11
y
12
y
13
y
14
y
15
y
0
, y
4
, y
8
, y
12
, y
1
, y
5
, y
9
, y
13
, . . .
y
n
m × mg
δ = 1
o
360
Np
Np
δ = 3
o
E
b
= E
C
=
mg+1
i=1
Pr(s
i
)E
s
i
s
i
s
i
∈ {s
1
, s
2
, . . . , s
M
} ∠s
i
i = M
∠s
i
δ
∠s
i
≥ 2π
∠s
i
= 0
o
i = i − 1
i = 0
mg+1
i=1
Pr(s
i
)E
s
i
= 1
r
n
(t) = α
n
(t)s
n
p
n
(t) + n
n
(t), nT
s
≤ t ≤ (n + 1)T
s
,
s
n
y
n
p(t)
α
n
(t) t
[nT
s
, (n + 1)T
s
) n
n
(t)
α(t)
G( ) =
1
1−
D
2
,
| | <
D
0,
| | ≥
D
D
r
n
(t)
r
n
= r
n
f
+ jr
n
q
r
n
= α
n
s
n
+ n
n
s
n
= s
n
f
+ js
n
q
[nT
s
, (n +
1)T
s
)
n
n
ℵ
0
/2
ℵ
0
= 10
−0,1(
E
b
ℵ
0
)
dB
.
ˆα(n)
ˆα(n + 1) = ˆα(n) + µs(n)e
∗
(n)
µ e(n) = r
n
− ˆα
n
ˆs(n)
ˆ
s
n
ˆ
¯y
n
ˆy
n
mg + 1
mg
mg +1
ˆ
s
n
ˆ
¯y
n
2 × 128
2 × 128
42
0
258
0
138
0
120
0
0
4
−4
−16
16
22
−22
28
−28
r = 1,59 r = 0,7
10
−10
2×128
4
0
27
0
60
0
125
0
110
−4
0
10
1622
28
−10
−16
−22
−28
2×128
2 × 128
2 × 8 2 × 128
0
4
84
0
117
0
102
0
2
−2
−4
6
8
−8
−6
r = 0,6r = 1,604
2 × 8
0
4
2
−2
8
−8
0
42
0
102
0
87
0
66
−6
−4
6
2 × 8
2 × 8
2 × 8 2 × 128
f
D
T
s
= 0, 002
f
D
T
s
= 0, 002
2 × 8
2 × 128
2 × 8 2 × 128
mg
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
MCW-PSK 2 x 8 Sem Entrel.
MCW-PSK 2 x 8 Prof. Entrel. 3
MCW-PSK 2 x 8 Prof. Entrel. 6
MCW-PSK 2 x 8 Prof. Entrel. 10
MCW-PSK 2 x 8 Doppler Inf.
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
MCW-PSK 2 x 128 Sem Entrel.
MCW-PSK 2 x 128 Prof. Entrel. 5
MCW-PSK 2 x 128 Prof. Entrel. 40
MCW-PSK 2 x 128 Prof. Entrel. 130
MCW-PSK 2 x 128 Doppler Inf.
2 ×128
2 ×8 2 ×128 f
D
T
s
= 0, 002
mg
mg
f
D
T
s
2 × 8 2 × 128
f
D
T
s
= 0, 002
•
•
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
MCW-APK 2 x 8 Sem Entrel.
MCW-APK 2 x 8 Prof. Entrel. 4
MCW-APK 2 x 8 Prof. Entrel. 7
MCW-APK 2 x 8 Prof. Entrel. 10
MCW-APK 2 x 8 Doppler Inf.
2 × 8
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
MCW-APK 2 x 128 Sem Entrel.
MCW-APK 2 x 128 Prof. Entrel. 20
MCW-APK 2 x 128 Prof. Entrel. 60
MCW-APK 2 x 128 Prof. Entrel. 130
MCW-APK 2 x 128 Doppler Inf.
2 ×128
2 ×8 2 ×128 f
D
T
s
= 0, 002
•
t
c
i
t
i = 1, 2
t r
t
r
t
=
2
i=1
α
i
c
i
t
+ n
t
.
α
i
i
l
t=1
|r
t
−
2
i=1
α
i
c
i
t
|
2
2 × 128
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
5 10 15 20 25
BER
Eb/No (dB)
Espacio-Temporal
MCW-PSK 2 x 128
MCW-APK 2 x 128
2 ×128
10
−5
E
b
/ℵ
0
10
−4
10
−5
mg
m = 2 g = 64
mg
2 × 8
2 ×128
f
D
T
s
= 0, 002
f
D
T
s
= 0, 004
µ
f
D
T
s
µ E
b
/ℵ
0
E
b
/ℵ
0
µ
µ
µ
2×8 2×128 f
D
T
s
= 0, 002
µ
f
D
T
s
= 0, 004
1e-04
1e-03
1e-02
1e-01
0 5 10 15 20 25 30 35
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-PSK 2 x 8
MCW-PSK 2 x 128
2 ×8 µ = 0, 70 2 ×128
µ = 0, 80
1e-04
1e-03
1e-02
1e-01
0 5 10 15 20 25 30 35
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-APK 2 x 8
MCW-APK 2 x 128
2 ×8 µ = 0, 90 2 ×128
µ = 0, 85
2 ×8 2 ×128 f
D
T
s
= 0, 002
1e-04
1e-03
1e-02
1e-01
0 5 10 15 20 25 30 35
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-PSK 2 x 8
MCW-PSK 2 x 128
2 ×8 µ = 0, 90 2 ×128
µ = 0, 95
1e-03
1e-02
1e-01
0 5 10 15 20 25 30 35
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-APK 2 x 8
MCW-APK 2 x 128
2 ×8 µ = 0, 95 2 ×128
µ = 0, 85
2×8 2×128 f
D
T
s
= 0, 004
f
D
T
s
2 × 8 2 × 128
f
D
T
s
= 0, 002 f
D
T
s
= 0, 004
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 8 LMS
MCW-APK 2 x 8 LMS
MCW-PSK 2 x 8 CSI
MCW-APK 2 x 8 CSI
2 × 8
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 128 LMS
MCW-APK 2 x 128 LMS
MCW-PSK 2 x 128 CSI
MCW-APK 2 x 128 CSI
2 × 128
2 × 8
2 × 128 f
D
T
s
= 0, 002
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 8 LMS
MCW-APK 2 x 8 LMS
MCW-PSK 2 x 8 CSI
MCW-APK 2 x 8 CSI
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 128 LMS
MCW-APK 2 x 128 LMS
MCW-PSK 2 x 128 CSI
MCW-APK 2 x 128 CSI
2 × 128
2 × 8
2 × 128 f
D
T
s
= 0, 004
m × mg
m m
y
j
n
, 0 ≤ j < m n
[nT
s
, (n + 1)T
s
)
y
n
nT
s
/m
I
d
L
L × I
d
m
m y
j
n
m [nT
s
, (n + 1)T
s
)
Desentrelacador
LMS
Estimador
.
.
.
Entrelacador
m
m
m
D
0
D
−(m−1)
D
−1
S/P
Conv.
.
.
.
m−1
MCW
MCW
1
0
MCW
.
Demodulador
Modulador
Canal
Antena
Fonte
Receptor
Decodificador
Wavelet
m−1
j=0
s
j
n
ˆ
s
j
n
x
n
ˆx
n
y
0
n
y
1
n
y
m−1
n
¯y
j
n
ˆ
¯y
j
n
ˆy
n
y
j
n
ˆy
j
n
r
j
n
r
j
n
ˆα
n
¯y
j
n
g + 1
¯y
j
n
ℵ
0
/2
ℵ
0
= 10
−0,1(
E
b
ℵ
0
)
dB
.
m nT
s
1/m
ˆ
¯y
j
n
ˆy
j
n
ˆy
n
2 ×8
4 × 16
2 × 8 4 × 16
f
D
T
s
= 0, 002
0
94
4
0
−4
r = 1,545
r = 0,48
−2
2
2 × 8
0
2
−2
0
132
−4
4
r = 0,707
2 × 8
0
4
−4
r = 0,818 r = 0,436
−2
2
0
109
4 × 16
0
2
−2
0
132
−4
4
r = 0,5
4 × 16
2 × 8
4 ×16
2 × 8 4 × 16
f
D
T
s
= 0, 002
m
2
g
f
D
T
s
= 0, 002
m
m [nT
s
, (n + 1)T
s
)
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 2 x 8 Sem Entrel.
MCW-PSK 2 x 8 Prof. Entrel. 5
MCW-PSK 2 x 8 Prof. Entrel. 10
MCW-PSK 2 x 8 Prof. Entrel. 20
MCW-PSK 2 x 8 Doppler Inf.
2 × 8
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 4 x 16 Sem Entrel.
MCW-PSK 4 x 16 Prof. Entrel. 5
MCW-PSK 4 x 16 Prof. Entrel. 20
MCW-PSK 4 x 16 Prof. Entrel. 70
MCW-PSK 4 x 16 Doppler Inf.
4 × 16
2 × 8 4 × 16 f
D
T
s
= 0, 002
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-APK 2 x 8 Sem Entrel.
MCW-APK 2 x 8 Prof. Entrel. 5
MCW-APK 2 x 8 Prof. Entrel. 20
MCW-APK 2 x 8 Doppler Inf.
2 × 8
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-APK 4 x 16 Sem Entrel.
MCW-APK 4 x 16 Prof. Entrel. 5
MCW-APK 4 x 16 Prof. Entrel. 25
MCW-APK 4 x 16 Prof. Entrel. 70
MCW-APK 4 x 16 Doppler Inf.
4 × 16
2 × 8 4 × 16 f
D
T
s
= 0, 002
2 × 8 4 × 16
2 × 8
2 ×8
10
−4
4 ×16 E
b
/ℵ
0
•
•
m m
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 2 x 8 SD
MCW-PSK 2 x 8 Div
MCW-PSK 4 x 16 Div
2 × 8 4 × 16
f
D
T
s
= 0, 002
2 × 8
µ
f
D
T
s
µ
E
b
/ℵ
0
E
b
/ℵ
0
1e-04
1e-03
1e-02
1e-01
1e+00
1 11 21 31 41
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-PSK 2 x 8
µ = 1, 28
1e-03
1e-02
1e-01
1e+00
2 17 32
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-APK 2 x 8
µ = 1, 06
2 × 8 f
D
T
s
= 0, 002
2 × 8 4 × 16
f
D
T
s
= 0, 002 µ
1e-04
1e-03
1e-02
1e-01
1e+00
2 17 32
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-PSK 4 x 16
µ = 1, 35
1e-03
1e-02
1e-01
1e+00
2 17 32
Erro Medio Quadratico
Intervalo de Sinalizacao
MCW-APK 4 x 16
µ = 1, 46
4 × 16 f
D
T
s
= 0, 002
2 ×8 4 ×16 f
D
T
s
= 0, 002
2 ×8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 8 LMS
MCW-APK 2 x 8 LMS
MCW-PSK 2 x 8 CSI
MCW-APK 2 x 8 CSI
2 × 8
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 4 x 16 LMS
MCW-APK 4 x 16 LMS
MCW-PSK 4 x 16 CSI
MCW-APK 4 x 16 CSI
4 × 16
2 × 8 4 × 16 f
D
T
s
= 0, 002
8 ·10
−5
6 ·10
−6
4 ×16
2 ×8
4 · 10
−4
2 · 10
−6
4 × 16 2 × 8
m × mg mg + 1
mg
y
n
mg + 1 y
n
mg · T
s
ω
n
= y
n
+ n
n
y
n
n
n
ℵ
0
/2
z
i
= S
i
+ R
i
S
i
R
i
mg
mg
{n
n
} ℵ
0
/2
R
i
E[R
2
i
] = mg
ℵ
0
2
P
e
=
1
2
Pr(S
i
+ R
i
> 0 | x
i−(mg−1)
= −1) +
+
1
2
Pr(S
i
+ R
i
< 0 | x
i−(mg−1)
= +1)
=
1
2
Pr(R
i
> mg | x
i−(mg−1)
= −1) +
+
1
2
Pr(R
i
> mg | x
i−(mg−1)
= +1)
R
i
x
i−(mg−1)
P
e
P
e
= Pr(R
i
> mg)
= Q
2mg
ℵ
0
mg
P
e
= Q
2E
b
ℵ
0
E
b
y
n
n mg+1
y
n
∈ {−mg, −mg + 2, . . . , −mg + 2k, . . . , −2, 0, 2, . . . , mg − 2, mg}
[nT
s
, (n + 1)T
s
)
Codificador
Wavelet
Decodificador
Wavelet
Desentrelacador
Entrelacador
Modulador
Demodulador
Antena
Canal
Receptor
Fonte
s
n
x
n
˜x
n
y
n
¯y
n
˜
¯y
n
˜y
n
r
n
r
n
r
n
= α
n
s
n
+ n
n
s
n
y
n
α
n
[nT
s
, (n + 1)T
s
)
n
n
ℵ
0
/2
˜y
n
˜y
n
˜y
n
= y
n
+ e
n
e
n
a
j
i = m(g + p) − 1
z
j
i
=
mg−1
k=0
a
j
(mg−1)−k
˜y
i−k
= η
j
i
+ ν
j
i
η
j
i
ν
j
i
η
j
i
=
mg−1
k=0
a
j
(mg−1)−k
y
i−k
ν
j
i
=
mg−1
k=0
a
j
(mg−1)−k
e
i−k
,
ν
j
i
ν
j
i
∈ Z
z
j
i
x
j+i−(mg−1)
x
j+i−(mg−1)
= −1 z
j
i
< 0 x
j+i−(mg−1)
= +1 z
j
i
> 0 z
j
i
= 0
x
j+i−(mg−1)
P
e
= P
z
j
e
= Pr(η
j
i
+ ν
j
i
> 0 | x
j+i−(mg−1)
= −1) · Pr(x
j+i−(mg−1)
= −1) +
+ 0.5 Pr(η
j
i
+ ν
j
i
= 0 | x
j+i−(mg−1)
= −1) · Pr(x
j+i−(mg−1)
= −1) +
+ Pr(η
j
i
+ ν
j
i
< 0 | x
j+i−(mg−1)
= +1) · Pr(x
j+i−(mg−1)
= +1) +
+ 0.5 Pr(η
j
i
+ ν
j
i
= 0 | x
j+i−(mg−1)
= +1) · Pr(x
j+i−(mg−1)
= +1).
E
z
j
i
= E
η
j
i
+ E
ν
j
i
= mg ·sgn
x
j+i−(mg−1)
P
z
j
e
z
j
m z
j
z
0
z
i
:=
z
0
i
, η
i
:= η
0
i
ν
i
:= ν
0
i
η
i
= mgx
i−(mg−1)
Pr(ν
i
>
mg |x
i−(mg−1)
= −1) = Pr(ν
i
< −mg |x
i−(mg−1)
= +1)
Pr(ν
i
= mg |x
i−(mg−1)
= −1) =
Pr(ν
i
= −mg |x
i−(mg−1)
= +1)
P
e
= Pr(ν
i
> mg | x
i−(mg−1)
= −1)) + 0.5 Pr(ν
i
= mg | x
i−(mg−1)
= −1)
ν
i
e
n
2 × 8
{−8, −6, −4, −2, 0, 2, 4, 6, 8}
e
n
= 0
e
n
e
n
(y
n
, ˜y
n
)
(0, 2), (2, 4), (4, 6), (6, 8), (−2, 0), (−4, −2), (−6, −4), (−8, −6)
(0, −2), (2, 0), (4, 2), (6, 4), (8, 6), (−6, −8), (−4, −6), (−2, −4)
(0, 4), (2, 6), (4, 8), (−8, −4), (−6, −2), (−4, 0), (−2, 2)
(0, −4), (2, −2), (4, 0), (6, 2), (8, 4), (−4, −8), (−2, −6)
(0, 6), (2, 8), (−8, −2), (−6, 0), (−4, 2), (−2, 4)
(0, −6), (2, −4), (4, −2), (6, 0), (8, 2), (−2, −8)
(0, 8), (−8, 0), (−6, 2), (−4, 4), (−2, 6)
(0, −8), (8, 0), (2, −6), (4, −4), (6, −2)
(−8, 2), (−6, 4), (−4, 6), (−2, 8)
(2, −8), (4, −6), (6, −4), (8, −2)
(−8, 4), (−6, 6), (−4, 8)
(4, −8), (6, −6), (8, −4)
(−8, 6), (−6, 8)
(6, −8), (8, −6)
(−8, 8)
(8, −8)
{y
n
}
I(e
n
; e
n+1
) 2 ×8
e
n
m × mg
e
n
= e
Pr(e
n
= e | x
i−(mg−1)
) =
E(e)
Pr(˜y
n
| y
n
) · Pr(y
n
| x
i−(mg−1)
),
e ∈ {−2mg, . . . , −2k, . . . , 0, . . . , 2k, . . . , 2mg} E(e) = {(y
n
, ˜y
n
) : ˜y
n
− y
n
= e}
Pr(e
n
|x
i−(mg−1)
)
Pr(˜y
n
| y
n
) Pr(y
n
| x
i−(mg−1)
)
x
i−(mg−1)
y
n
2 × 8
i = m(g + p) − 1
p ∈ N z
i
x
i−(mg−1)
(y
i−(mg−1)
, y
i−(mg−2)
, . . . , y
i
) y
x
i−(mg−1)
y
x
8
x
8
2 × 8
x
8
= −1 y
x
8
1 1 1 −1 1 1 −1 1
1 1 1 −1 −1 −1 1 −1
.
mg −m
n y
n
a
0
0
x
0
+ a
1
0
x
1
a
0
1
x
0
+ a
1
1
x
1
a
0
2
x
0
+ a
1
2
x
1
+ a
0
0
x
2
+ a
1
0
x
3
a
0
3
x
0
+ a
1
3
x
1
+ a
0
1
x
2
+ a
1
1
x
3
a
0
4
x
0
+ a
1
4
x
1
+ a
0
2
x
2
+ a
1
2
x
3
+ a
0
0
x
4
+ a
1
0
x
5
a
0
5
x
0
+ a
1
5
x
1
+ a
0
3
x
2
+ a
1
3
x
3
+ a
0
1
x
4
+ a
1
1
x
5
a
0
6
x
0
+ a
1
6
x
1
+ a
0
4
x
2
+ a
1
4
x
3
+ a
0
2
x
4
+ a
1
2
x
5
+ a
0
0
x
6
+ a
1
0
x
7
a
0
7
x
0
+ a
1
7
x
1
+ a
0
5
x
2
+ a
1
5
x
3
+ a
0
3
x
4
+ a
1
3
x
5
+ a
0
1
x
6
+ a
1
1
x
7
a
0
6
x
2
+ a
1
6
x
3
+ a
0
4
x
4
+ a
1
4
x
5
+ a
0
2
x
6
+ a
1
2
x
7
+ a
0
0
x
8
+ a
1
0
x
9
a
0
7
x
2
+ a
1
7
x
3
+ a
0
5
x
4
+ a
1
5
x
5
+ a
0
3
x
6
+ a
1
3
x
7
+ a
0
1
x
8
+ a
1
1
x
9
a
0
6
x
4
+ a
1
6
x
5
+ a
0
4
x
6
+ a
1
4
x
7
+ a
0
2
x
8
+ a
1
2
x
9
+ a
0
0
x
10
+ a
1
0
x
11
a
0
7
x
4
+ a
1
7
x
5
+ a
0
5
x
6
+ a
1
5
x
7
+ a
0
3
x
8
+ a
1
3
x
9
+ a
0
1
x
10
+ a
1
1
x
11
a
0
6
x
6
+ a
1
6
x
7
+ a
0
4
x
8
+ a
1
4
x
9
+ a
0
2
x
10
+ a
1
2
x
11
+ a
0
0
x
12
+ a
1
0
x
13
a
0
7
x
6
+ a
1
7
x
7
+ a
0
5
x
8
+ a
1
5
x
9
+ a
0
3
x
10
+ a
1
3
x
11
+ a
0
1
x
12
+ a
1
1
x
13
a
0
6
x
8
+ a
1
6
x
9
+ a
0
4
x
10
+ a
1
4
x
11
+ a
0
2
x
12
+ a
1
2
x
13
+ a
0
0
x
14
+ a
1
0
x
15
a
0
7
x
8
+ a
1
7
x
9
+ a
0
5
x
10
+ a
1
5
x
11
+ a
0
3
x
12
+ a
1
3
x
13
+ a
0
1
x
14
+ a
1
1
x
15
y
x
8
n y
n
| (x
8
= −1)
−x
2
+ x
3
+ x
4
− x
5
+ x
6
+ x
7
+ x
9
− 1
x
2
− x
3
+ x
4
− x
5
− x
6
− x
7
+ x
9
− 1
−x
4
+ x
5
+ x
6
− x
7
+ x
9
+ x
10
+ x
11
− 1
x
4
− x
5
+ x
6
− x
7
− x
9
+ x
10
+ x
11
+ 1
−x
6
+ x
7
− x
9
+ x
10
+ x
11
+ x
12
+ x
13
− 1
x
6
− x
7
− x
9
− x
10
− x
11
+ x
12
+ x
13
− 1
x
9
+ x
10
− x
11
+ x
12
+ x
13
+ x
14
+ x
15
+ 1
−x
9
+ x
10
− x
11
− x
12
− x
13
+ x
14
+ x
15
− 1
y
n
x
i−(mg−1)
= −1
y
n
=
mg−1
k=1
b
k
x
k
+ l
{x
k
} mg − 1
y
n
b
k
x
k
±1 l = l(a
s
k
) = −a
s
k
y
x
8
l(a
s
k
) = +1
x
i−(mg−1)
= −1
a
0
3
a
0
6
y
n
G
y
n
| x
i−(mg−1)
=−1
(z) = E
z
mg−1
k=1
b
k
x
k
+l
= z
l
mg−1
k=1
E
z
b
k
x
k
= 0.5
mg−1
z
l
z + z
−1
mg−1
=
mg−1
k=0
mg − 1
k
z
2k−mg+l+1
0.5
mg−1
x
k
∈ {+1, −1}
x
k
Pr(y
n
= 2k − mg + l + 1 | x
i−(mg−1)
= −1) =
mg − 1
k
0.5
mg−1
, 0 ≤ k ≤ mg − 1,
l ∈ {±1}
y
n
x
i−(mg−1)
= −1
n l = −1
l = +1 (mg = 8)
l = −1 l = +1
Pr(y
n
= −8 | x
i−(mg−1)
= −1) = 0.5
7
Pr(y
n
= −8 | x
i−(mg−1)
= −1) = 0.0
Pr(y
n
= −6 | x
i−(mg−1)
= −1) = 7 · 0.5
7
Pr(y
n
= −6 | x
i−(mg−1)
= −1) = 0.5
7
Pr(y
n
= −4 | x
i−(mg−1)
= −1) = 21 · 0.5
7
Pr(y
n
= −4 | x
i−(mg−1)
= −1) = 7 · 0.5
7
Pr(y
n
= −2 | x
i−(mg−1)
= −1) = 35 · 0.5
7
Pr(y
n
= −2 | x
i−(mg−1)
= −1) = 21 · 0.5
7
Pr(y
n
= 0 | x
i−(mg−1)
= −1) = 35 · 0.5
7
Pr(y
n
= 0 | x
i−(mg−1)
= −1) = 35 · 0.5
7
Pr(y
n
= 2 | x
i−(mg−1)
= −1) = 21 · 0.5
7
Pr(y
n
= 2 | x
i−(mg−1)
= −1) = 35 · 0.5
7
Pr(y
n
= 4 | x
i−(mg−1)
= −1) = 7 · 0.5
7
Pr(y
n
= 4 | x
i−(mg−1)
= −1) = 21 · 0.5
7
Pr(y
n
= 6 | x
i−(mg−1)
= −1) = 0.5
7
Pr(y
n
= 6 | x
i−(mg−1)
= −1) = 7 · 0.5
7
Pr(y
n
= 8 | x
i−(mg−1)
= −1) = 0.0
Pr(y
n
= 8 | x
i−(mg−1)
= −1) = 0.5
7
Pr
l=−1
(e
n
= e | x
i−(mg−1)
= −1) = Pr
l=+1
(e
n
= −e | x
i−(mg−1)
= −1).
l(a
s
k
)
Pr(e
n
| x
i−(mg−1)
)
Pr(˜y
n
|y
n
)
2 × 8 2 × 128
Pr(˜y
n
|y
n
)
Pr(˜y
n
|y
n
)
Pr(˜y
n
|y
n
) =
2
πℵ
0
∞
0
Θ
˜y
n
∞
0
αV · exp
−
V
2
− 2αV
√
E
b
cos(Θ
s
− Θ
r
) + α
2
(E
b
+ ℵ
0
)
ℵ
0
· dV dΘ
r
dα
α s
n
f
s
n
q
y
n
r
n
f
r
n
q
r
n
Θ
˜y
n
˜y
n
Θ
s
= tan
−1
(s
n
q
/s
n
f
)
Θ
r
= tan
−1
(r
n
q
/r
n
f
)
V =
r
2
n
f
+ r
2
n
q
E
b
= E[α
2
] = 1
e
n
ν
i
ν
i
x
i−(mg−1)
= −1
G
(ν
i
| x
i−(mg−1)
=−1)
(z) = E
z
ν
i
| x
i−(mg−1)
= −1
= E
z
mg−1
k=0
a
0
(mg−1)−k
e
i−k
| x
i−(mg−1)
= −1
= E
mg−1
k=0
z
a
0
(mg−1)−k
e
i−k
| x
i−(mg−1)
= −1
{e
n
}
G
(ν
i
| x
i−(mg−1)
=−1)
(z) =
mg−1
k=0
E
z
a
0
(mg−1)−k
e
i−k
| x
i−(mg−1)
= −1
e
n
x
i−(mg−1)
a
s
k
x
i−(mg−1)
a
s
k
G
(ν
i
| x
i−(mg−1)
=−1)
(z) =
mg−1
k=0
E
z
a
0
(mg−1)−k
e
i−k
| x
i−(mg−1)
= −1
=
k: a
0
(mg−1)−k
=+1
E
z
e
i−k
| x
i−(mg−1)
·
k: a
0
(mg−1)−k
=−1
E
z
−e
i−k
| x
i−(mg−1)
E
z
e
i−k
|x
i−(mg−1)
= −1
a
k
=+1
= E
z
−e
i−k
|x
i−(mg−1)
= −1
a
k
=−1
G
(ν
i
|x
i−(mg−1)
=−1)
(z) =
E
z
e
| x
i−(mg−1)
= −1
mg
E
z
ν
i
|x
i−(mg−1)
= −1
=
E
z
e
|x
i−(mg−1)
= −1
mg
P
e
=
mg(2mg−1)
2
k=1
Pr(ν
i
= mg + 2k |x
i−(mg−1)
= −1) + 0.5 Pr(ν
i
= mg |x
i−(mg−1)
= −1)
2 ×128
2 × 128
2 ×128
e
n
y
n
∈ {2, 4, 6}
e
k
y
n
nT
s
e
k
e
k
{−2, 0, 2} (y
k
= 4, ˜y
k
= 4)
e
q
k
e
k
e
q
k
2 × 128
2 × 128
e
q
k
Pr(y
n
) · Pr(y
k
→ y
k
)
Pr(2) · Pr(4 → 4) + Pr(8) · Pr(10 → 10) + Pr(14) · Pr(16 → 16) + Pr(20) · Pr(22 → 22)+
+ Pr(26) ·Pr(28 → 28) + Pr(−30) · Pr(−28 → −28) + Pr(−24) · Pr(−22 → −22)+
+ Pr(−18) ·Pr(−16 → −16) + Pr(−12) · Pr(−10 → −10) + Pr(−6) · Pr(−4 → −4)
Pr(6) · Pr(4 → 4) + Pr(12) · Pr(10 → 10) + Pr(18) · Pr(16 → 16) + Pr(24) · Pr(22 → 22)+
+ Pr(30) ·Pr(28 → 28) + Pr(−26) · Pr(−28 → −28) + Pr(−20) · Pr(−22 → −22)+
+ Pr(−14) ·Pr(−16 → −16) + Pr(−8) · Pr(−10 → −10) + Pr(−2) · Pr(−4 → −4)
Pr(−32) · Pr(−28 → −28)
Pr(32) · Pr(28 → 28)
Pr(−34) · Pr(−28 → −28)
Pr(34) · Pr(28 → 28)
Pr(−128) · Pr(−28 → −28)
Pr(128) · Pr(28 → 28)
p(4 28)
p(4 28)
p(4 22)
p(4 22)
p(4 22)
p(2)
p(4)
p(6)
p(2)
p(4)
p(6)
p(4 16)
p(4 16)
p(4 16)
p(2)
p(4)
p(6)
p(4 10)
p(4 10)
p(4 10)
p(2)
p(4)
p(6)
p(2)
p(4)
p(6)
p(4 0)
p(4 0)
p(4 0)
p(4 4)
p(4 4)
p(4 4)
p(2)
p(4)
p(6)
p(4 −4)
p(4 −4)
p(4 −4)
p(2)
p(4)
p(6) p(4 −10)
p(4 −10)
p(4 −10)
p(2)
p(4)
p(6)
p(4 −16)
p(4 −16)
p(4 −16)
p(2)
p(4)
p(6)
p(4 −22)
p(4 −22)
p(4 −22)
p(2)
p(4)
p(6)
p(4 −28)
p(4 −28)
p(4 −28)
e = 26 p(2) p(4 28)
e = 24 p(4)
e = 22 p(6)
28
e = 18
e = 20
e = 16
22
e = 14
e = 12
e = 10
16
e = 8
e = 6
e = 4
10
e = 2
e = 0
e = −2
4
0
e = −10
e = −8
e = −6
e = −4
e = −6
e = −2
−4
e = −12
e = −14
e = −16
−10
e = −20
e = −18
e = −22
−16
e = −24
e = −26
e = −28
−22
e = −30
e = −32
e = −34
−28
4
2
4
6
{2, 4, 6}
2 × 128
e
q
k
2, 13 · 10
−10
2 × 8
2 × 128
mg
2 × 8
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 8 Simul. SD
MCW-PSK 2 x 8 Analit. SD
MCW-APK 2 x 8 Simul. SD
MCW-APK 2 x 8 Analit. SD
2 × 8
{e
n
}
2 × 128
2 ×128
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW-PSK 2 x 8 Simul. SD
MCW-PSK 2 x 8 Analit. SD
MCW-APK 2 x 8 Simul. SD
MCW-APK 2 x 8 Analit. SD
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
MCW-PSK 2 x 128 Simul. SD
MCW-PSK 2 x 128 Analit. SD
MCW-APK 2 x 128 Simul. SD
MCW-APK 2 x 128 Analit. SD
2 × 128
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
Space-Time
MCW-PSK 2 x 128 Simul. SD
MCW-PSK 2 x 128 Analit. SD
MCW-APK 2 x 128 Simul. SD
MCW-APK 2 x 128 Analit. SD
2 × 128
10
−5
2×128
m
y
j
n
n
g + 1
y
j
n
∈ {−g, −g + 2, . . . , −g + 2k, . . . , −2, 0, 2, . . . , g − 2, g}
[nT
s
, (n+1)T
s
) m
Entrelacador
m
m
m
D
0
D
−(m−1)
D
−1
S/P
Conv.
.
.
.
m−1
MCW
MCW
1
0
MCW
Desentrelacador
Decodificador
Wavelet
Receptor
Modulador
Canal
Antena
Fonte
Demodulador
m−1
j=0
s
j
n
x
n
˜x
n
y
0
n
y
1
n
y
m−1
n
¯y
j
n
˜
¯y
j
n
˜y
n
y
j
n
˜y
j
n
r
j
n
m
r
j
n
= α
n
s
j
n
+ n
n
, 0 ≤ j ≤ m − 1
s
j
n
¯y
j
n
nT
s
α
n
n
n
ℵ
0
/2
[nT
s
, (n+1)T
s
) m
˜
¯y
j
n
m ˜y
j
n
(0 ≤ j ≤ m − 1)
˜y
n
˜y
n
˜y
n
=
m−1
j=0
˜y
j
n
=
m−1
j=0
(y
j
n
+ e
j
n
)
e
j
n
a
0
i = m(g + p) − 1
z
i
= η
i
+ ν
i
,
η
i
ν
i
η
i
=
mg−1
k=0
m−1
j=0
a
0
(mg−1)−k
y
j
i−k
ν
i
=
mg−1
k=0
m−1
j=0
a
0
(mg−1)−k
e
j
i−k
P
e
= Pr(ν
i
> mg | x
i−(mg−1)
= −1) + 0.5 Pr(ν
i
= mg | x
i−(mg−1)
= −1)
e
j
n
x
i−(mg−1)
Pr(e
j
n
= e | x
i−(mg−1)
) =
E
j
(e)
Pr(˜y
j
n
| y
j
n
) · Pr(y
j
n
| x
i−(mg−1)
),
e
j
n
∈ {−2g, . . . , −2k, . . . , 0, . . . , 2k, . . . , 2g}
E
j
(e) = {(y
j
n
, ˜y
j
n
) : ˜y
j
n
− y
j
n
= e}
2 × 8
1 1 1 −1 1 1 −1 1
1 1 1 −1 −1 −1 1 −1
.
nT
s
y
0
n
y
1
n
a
0
0
x
0
a
1
0
x
1
a
0
1
x
0
a
1
1
x
1
a
0
2
x
0
+ a
0
0
x
2
a
1
2
x
1
+ a
1
0
x
3
a
0
3
x
0
+ a
0
1
x
2
a
1
3
x
1
+ a
1
1
x
3
a
0
4
x
0
+ a
0
2
x
2
+ a
0
0
x
4
a
1
4
x
1
+ a
1
2
x
3
+ a
1
0
x
5
a
0
5
x
0
+ a
0
3
x
2
+ a
0
1
x
4
a
1
5
x
1
+ a
1
3
x
3
+ a
1
1
x
5
a
0
6
x
0
+ a
0
4
x
2
+ a
0
2
x
4
+ a
0
0
x
6
a
1
6
x
1
+ a
1
4
x
3
+ a
1
2
x
5
+ a
1
0
x
7
a
0
7
x
0
+ a
0
5
x
2
+ a
0
3
x
4
+ a
0
1
x
6
a
1
7
x
1
+ a
1
5
x
3
+ a
1
3
x
5
+ a
1
1
x
7
a
0
6
x
2
+ a
0
4
x
4
+ a
0
2
x
6
+ a
0
0
x
8
a
1
6
x
3
+ a
1
4
x
5
+ a
1
2
x
7
+ a
1
0
x
9
a
0
7
x
2
+ a
0
5
x
4
+ a
0
3
x
6
+ a
0
1
x
8
a
1
7
x
3
+ a
1
5
x
5
+ a
1
3
x
7
+ a
1
1
x
9
a
0
6
x
4
+ a
0
4
x
6
+ a
0
2
x
8
+ a
0
0
x
10
a
1
6
x
5
+ a
1
4
x
7
+ a
1
2
x
9
+ a
1
0
x
11
a
0
7
x
4
+ a
0
5
x
6
+ a
0
3
x
8
+ a
0
1
x
10
a
1
7
x
5
+ a
1
5
x
7
+ a
1
3
x
9
+ a
1
1
x
11
a
0
6
x
6
+ a
0
4
x
8
+ a
0
2
x
10
+ a
0
0
x
12
a
1
6
x
7
+ a
1
4
x
9
+ a
1
2
x
11
+ a
1
0
x
13
a
0
7
x
6
+ a
0
5
x
8
+ a
0
3
x
10
+ a
0
1
x
12
a
1
7
x
7
+ a
1
5
x
9
+ a
1
3
x
11
+ a
1
1
x
13
a
0
6
x
8
+ a
0
4
x
10
+ a
0
2
x
12
+ a
0
0
x
14
a
1
6
x
9
+ a
1
4
x
11
+ a
1
2
x
13
+ a
1
0
x
15
a
0
7
x
8
+ a
0
5
x
10
+ a
0
3
x
12
+ a
0
1
x
14
a
1
7
x
9
+ a
1
5
x
11
+ a
1
3
x
13
+ a
1
1
x
15
y
0
n
a
0
y
1
n
a
1
i = m(g + p) − 1
p ∈ N z
i
x
i−(mg−1)
(y
0
i−(mg−1)
, y
1
i−(mg−1)
, y
0
i−(mg−2)
, y
1
i−(mg−2)
, . . . , y
0
i
, y
1
i
)
(y
0
x
i−(mg−1)
, y
1
x
i−(mg−1)
) (y
0
x
8
, y
1
x
8
)
x
8
y
j
n
, j ∈ {0, . . . , m − 1} n
x
8
a
0
y
1
x
8
2 × 8 x
8
= −1
y
0
x
8
y
0
x
8
n y
0
n
|(x
8
= −1)
−x
2
+ x
4
+ x
6
− 1
x
2
+ x
4
− x
6
− 1
−x
4
+ x
6
+ x
10
− 1
x
4
+ x
6
+ x
10
+ 1
−x
6
+ x
10
+ x
12
− 1
x
6
− x
10
+ x
12
− 1
x
10
+ x
12
+ x
14
+ 1
x
10
− x
12
+ x
14
− 1
y
0
n
x
i−(mg−1)
= −1
y
0
n
=
g−1
k=1
b
k
x
k
+ l,
l = l(a
s
k
) = −a
s
k
Pr(y
0
n
= 2k − g + l + 1 | x
i−(mg−1)
= −1) =
g − 1
k
0.5
g−1
, 0 ≤ k ≤ g − 1,
l ∈ {±1}.
y
0
n
x
i−(mg−1)
= −1
n l = −1
l = +1 (g = 4)
l = −1 l = +1
Pr(y
0
n
= −4 | x
i−(mg−1)
= −1) = 0, 5
3
Pr(y
0
n
= −4 | x
i−(mg−1)
= −1) = 0, 0
Pr(y
0
n
= −2 | x
i−(mg−1)
= −1) = 3 · 0, 5
3
Pr(y
0
n
= −2 | x
i−(mg−1)
= −1) = 0, 5
3
Pr(y
0
n
= 0 | x
i−(mg−1)
= −1) = 3 · 0, 5
3
Pr(y
0
n
= 0 | x
i−(mg−1)
= −1) = 3 · 0, 5
3
Pr(y
0
n
= 2 | x
i−(mg−1)
= −1) = 0, 5
3
Pr(y
0
n
= 2 | x
i−(mg−1)
= −1) = 3 · 0, 5
3
Pr(y
0
n
= 4 | x
i−(mg−1)
= −1) = 0, 0
Pr(y
0
n
= 4 | x
i−(mg−1)
= −1) = 0, 5
3
Pr
l=−1
(e
0
n
= e | x
i−(mg−1)
= −1) = Pr
l=+1
(e
0
n
= −e | x
i−(mg−1)
= −1).
y
j
n
x
i−(mg−1)
= −1
y
j
n
x
i−(mg−1)
y
j
n
=
g
k=1
b
j
k
x
k
,
x
k
b
j
k
∈ {−1, +1}
y
1
x
8
y
j
n
G
y
j
n
(z) = E[z
g
k=1
x
k
b
j
k
] = E[z
x
]
g
= (
1
2
z +
1
2
z
−1
)
g
=
g
k=0
g
k
z
2k−g
· 0.5
g
y
j
n
x
i−(mg−1)
Pr(y
j
n
= 2k − g) =
g
k
0.5
g
, 0 ≤ k ≤ g
Pr(y
1
n
= −4) = 0.5
4
Pr(y
1
n
= −2) = 4 · 0.5
4
Pr(y
1
n
= 0) = 6 · 0.5
4
Pr(y
1
n
= 2) = 4 · 0.5
4
Pr(y
1
n
= 4) = 0.5
4
˜y
j
n
y
j
n
Pr(˜y
j
n
| y
j
n
) =
2
πℵ
0
∞
0
Θ
˜y
j
n
∞
0
αV · exp
−
V
2
− 2αV
√
E
s
cos(Θ
s
− Θ
r
) + α
2
(E
s
+ ℵ
0
)
ℵ
0
· dV dΘ
r
dα
α s
j
n
(p) s
j
n
(q)
y
j
n
Θ
s
= tan
−1
(s
j
n
(q)/s
j
n
(p))
Θ
˜y
j
n
˜y
j
n
V =
r
j
n
(p)
2
+ r
j
n
(q)
2
Θ
r
= tan
−1
(r
j
n
(q)/r
j
n
(p))
E
s
= 1/m
Pr(˜y
j
n
| y
j
n
)
e
j
n
ν
i
x
i−(mg−1)
= −1
G
(ν
i
| x
i−(mg−1)
=−1)
(z) = E
z
ν
i
| x
i−(mg−1)
= −1
= E
z
mg−1
k=0
m−1
j=0
a
0
(mg−1)−k
e
j
i−k
| x
i−(mg−1)
= −1
{e
j
n
}
nT
s
G
(ν
i
| x
i−(mg−1)
=−1)
(z) =
mg−1
k=0
E
m−1
j=0
z
a
0
(mg−1)−k
e
j
i−k
| x
i−(mg−1)
= −1
m y
j
k
, ∀j ∈
{0, . . . , m − 1}
nT
s
x
i−(mg−1)
e
j
n
y
j
n
G
(ν
i
| x
i−(mg−1)
=−1)
(z) =
mg−1
k=0
E
z
a
0
(mg−1)−k
e
j
i−k
m−1
j=0
· E
z
a
0
(mg−1)−k
e
0
i−k
| x
i−(mg−1)=−1
e
j
n
x
i−(mg−1)
e
0
n
x
i−(mg−1)
x
i−(mg−1)
a
s
k
x
i−(mg−1)
a
s
k
G
(ν
i
| x
i−(mg−1)
=−1)
(z) = E
z
e
j
n
mg(m−1)
j=0
·
k: a
0
(mg−1)−k
=+1
E
z
+e
0
i−k
| x
i−(mg−1)
= −1
·
·
k: a
0
(mg−1)−k
=−1
E
z
−e
0
i−k
| x
i−(mg−1)
= −1
Pr
l=−1
(e
0
n
= e | x
i−(mg−1)
= −1) = Pr
l=+1
(e
0
n
= −e | x
i−(mg−1)
= −1)
E
z
+e
0
i−k
| x
i−(mg−1)
= −1
a
k
=+1
= E
z
−e
0
i−k
| x
i−(mg−1)
= −1
a
k
=−1
G
(ν
i
| x
i−(mg−1)
=−1)
(z) = E
z
e
j
n
mg(m−1)
j=0
· E
z
e
0
n
| x
i−(mg−1)
= −1
mg
E
z
ν
i
| x
i−(mg−1)
= −1
= E
z
e
j
n
mg(m−1)
j=0
· E
z
e
0
n
| x
i−(mg−1)
= −1
mg
P
e
=
mg(2mg−1)
2
k=1
Pr(ν
i
= mg + 2k |x
i−(mg−1)
= −1) + 0.5 Pr(ν
i
= mg |x
i−(mg−1)
= −1)
2 ×8 4 ×16
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 2 x 8 Simul. Div.
MCW-PSK 2 x 8 Analit. Div.
MCW-APK 2 x 8 Simul. Div.
MCW-APK 2 x 8 Analit. Div.
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 4 x 16 Simul. Div.
MCW-PSK 4 x 16 Analit. Div.
MCW-APK 4 x 16 Simul. Div.
MCW-APK 4 x 16 Analit. Div.
4 × 16
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 2 x 8 Simul. Div.
MCW-PSK 2 x 8 Analit. Div.
MCW-APK 2 x 8 Simul. Div.
MCW-APK 2 x 8 Analit. Div.
2 × 8
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW-PSK 4 x 16 Simul. Div.
MCW-PSK 4 x 16 Analit. Div.
MCW-APK 4 x 16 Simul. Div.
MCW-APK 4 x 16 Analit. Div.
4 × 16
MATLAB
1
o
180
mg+1
N
r
· 180
mg+1
N
r
x = (x
1
, x
2
, . . . , x
k
) ∈ R
k
U(a
i
, b
i
) a
i
b
i
x
i
x
k
l = (a
1
, b
1
, a
2
, b
2
, . . . , a
k
, b
k
)
P
e
=
mg(2mg−1)
2
k=1
Pr(ν
i
= mg + 2k |x
i−(mg−1)
= −1) + 0.5 Pr(ν
i
= mg |x
i−(mg−1)
= −1)
Pr(ν
i
|x
i−(mg−1)
= −1) ν
i
x
i−(mg−1)
= −1 m g
F = −P
e
Pr(ν
i
|x
i−(mg−1)
= −1)
MATLAB
q
t
b
p
j
p
j
= q
′
(1 − q)
r
j
−1
q ∈ (0.0, 1.0)
r
j
j
r
j
= 1 q
′
=
q
1−(1−q)
P
p
j
P
k x y
x y
x
y x
′
x
′
= x + r(x − y)
r U(0, 1)
x
′
t
t = 3
x
′
= y
y
′
= x
a
i
b
i
x
i
x
x
x
′
i
=
x
i
+ (b
i
− x
i
)f(G)
r
1
< 0, 5,
x
i
− (x
i
+ a
i
)f(G) r
1
≥ 0, 5,
i ∈ {1, . . . , k} k x
f(G) =
r
2
1 −
G
Gmax
b
.
r
1
r
2
(0, 1) G
Gmax
b f(G)
b = 3 f(G)
f(G) G Gmax
2 × 8 2 × 128
2 × 128
2
−2
0
−4
4
0
78
0
88
0
104
r = 0,61r = 1,595
8
−6
6
−8
2 × 8
0
8
−8
6
−6
4
−4
2
−2
0
103
0
137
0
49
0
76
2 × 8
97
0
0
4
−4
r = 1,59 r = 0,7
0
10
−10
16
−16
0
0
28
22
−22
−28
134
33
124
2 × 128
4
0
0
0
0
0
−4
0
10
22
28
−10
−16
−22
−28
16
91
127
32
64
112
2 × 128
2 ×8 2 ×128
2 × 8
±6 ±8
2 × 128
mg + 1
2 × 8
δ = 3
o
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW 2 x 8 PSK EUC
MCW 2 x 8 PSK-AG EUC
MCW 2 x 8 APK EUC
MCW 2 x 8 APK-AG EUC
2 × 8
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW 2 x 128 PSK EUC
MCW 2 x 128 PSK-AG EUC
MCW 2 x 128 APK EUC
MCW 2 x 128 APK-AG EUC
2 ×128
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30 35
BER
Eb/No (dB)
MCW 2 x 8 PSK MAP
MCW 2 x 8 PSK-AG MAP
MCW 2 x 8 APK MAP
MCW 2 x 8 APK-AG MAP
2 × 8
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25
BER
Eb/No (dB)
MCW 2 x 128 PSK MAP
MCW 2 x 128 PSK-AG MAP
MCW 2 x 128 APK MAP
MCW 2 x 128 APK-AG MAP
2 × 128
2 × 8 4 × 16
2
−2
0
107
0
101
0
−4
4
r = 1,372 r = 0,55
2 × 8
0
2
−2
0
122
0
−4
4
89
r = 0,707
2 × 8
4
−4
0
104
85
0
0
r = 0,93 r = 0,4
2
−2
4 × 16
0
2
−2
0
127
0
88
−4
4
r = 0,5
4 × 16
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW 2 x 8 PSK EUC
MCW 2 x 8 PSK-AG EUC
MCW 2 x 8 APK EUC
MCW 2 x 8 APK-AG EUC
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW 2 x 8 PSK MAP
MCW 2 x 8 PSK-AG MAP
MCW 2 x 8 APK MAP
MCW 2 x 8 APK-AG MAP
2 × 8
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW 4 x 16 PSK EUC
MCW 4 x 16 PSK-AG EUC
MCW 4 x 16 APK EUC
MCW 4 x 16 APK-AG EUC
4 × 16
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 5 10 15 20 25 30
BER
Eb/No (dB)
MCW 4 x 16 PSK MAP
MCW 4 x 16 PSK-AG MAP
MCW 4 x 16 APK MAP
MCW 4 x 16 APK-AG MAP
4 × 16
2 × 128
mg m g
f
D
T
s
= 0, 002
m
2
g
N
Total
= 100 ·
1
N
Total
= L · N
Bloco
L = 10
1
1/2
N
Bloco
= 100
1
1/2
k = log
2
M
M = 2
k
s
m
(t) = g(t) cos(2πf
c
t + θ
m
),
g(t) [nT
s
, (n + 1)T
s
)
θ
m
=
2π(m − 1)
M
, m = 1, ..., M
M
s
m
=
E
s
cos
2π
M
(m − 1)
E
s
sin
2π
M
(m − 1)
E
s
=
1
2
E
g
C(r, s
m
) = r · s
m
, m = 1, 2, . . . , M
s
m
r = [r
f
r
q
] M s
m
s
m
Θ
r
= tan
−1
r
q
r
f
Θ
r
s = [s
f
s
q
]
r
f
= s
f
+ n
f
r
q
= s
q
+ n
q
n
f
n
q
r
f
r
q
E[r
f
] = s
f
E[r
q
] = s
q
σ
2
r
f
= σ
2
r
q
=
1
2
ℵ
0
= σ
2
r
p
r
(r
f
, r
q
) =
1
2πσ
2
r
exp
−
(r
f
− s
f
)
2
+ (r
q
− s
q
)
2
2σ
2
r
Θ
r
(r
f
, r
q
)
V =
r
2
f
+ r
2
q
Θ
r
= tan
−1
(r
q
/r
f
)
s
m
p
V,Θ
r
(V, Θ
r
) =
1
πℵ
0
exp
−
V
2
− 2V
√
E
s
cos(Θ
r
− Θ
m
) + E
s
ℵ
0
p
V,Θ
r
(V, Θ
r
)
V p
Θ
r
(Θ
r
)
p
Θ
r
(Θ
r
) =
∞
0
p
V,Θ
r
(V, Θ
r
)V dV
=
1
πℵ
0
∞
0
V exp
−
V
2
− 2V
√
E
s
cos(Θ
r
− Θ
m
) + E
s
ℵ
0
dV
s
1
−π/M ≤ Θ
r
≤ π/M
P
M
= 1 −
π/M
−π/M
p
Θ
r
(Θ
r
)dΘ
r
p
Θ
r
(Θ
r
)
M = 2 M = 4
M = 2
P
erro
= Q
2E
b
ℵ
0
Q(α) : =
1
√
2π
∞
α
e
−y
2
/2
dy
M = 4
M = 4
P
c
= (1 − P
2
)
2
=
1 − Q
2E
b
ℵ
0
2
P
c
M = 4
P
4
= 1 − P
c
= 2Q
2E
b
ℵ
0
1 −
1
2
Q
2E
b
ℵ
0
M > 4 P
M
p
Θ
r
(Θ
r
)
M E
s
/ℵ
0
>> 1 |Θ
r
| ≤
1
2
π
p
Θ
r
(Θ
r
)
p
Θ
r
(Θ
r
) ≈
E
b
πℵ
0
cos(Θ
r
− Θ
m
)e
−
E
b
ℵ
0
sin
2
(Θ
r
−Θ
m
)
p
Θ
r
(Θ
r
)
Θ
r
u =
E
b
ℵ
0
sin(Θ
r
− Θ
m
)
P
M
≈ 1 −
θ
2
θ
1
E
b
πℵ
0
cos(Θ
r
− Θ
m
)e
−
E
b
ℵ
0
sin
2
(Θ
r
−Θ
m
)
dΘ
r
≈ 1 −
1
√
π
√
E
b
/ℵ
0
sin(Θ
2
−Θ
m
)
√
E
b
/ℵ
0
sin(Θ
1
−Θ
m
)
e
−u
2
du
≈ 1 −
1
2
E
b
/ℵ
0
sin(Θ
2
− Θ
m
)
−
E
b
/ℵ
0
sin(Θ
1
− Θ
m
)
θ
1
θ
2
2
k
P
b
≈
1
k
P
M
s
m
(t)
r(t) = αe
−jφ
s
m
(t) + n(t), 0 ≤ nT
s
≤ T
n(t)
φ
α
α
[s
f
s
q
] α
r
f
= αs
f
+ n
f
r
q
= αs
q
+ n
q
r
f
r
q
p
r
(r
f
, r
q
|α) =
1
2πσ
2
r
exp
−
(r
f
− αs
f
)
2
+ (r
q
− αs
q
)
2
2σ
2
r
p
V,Θ
r
(V, Θ
r
|α) =
1
πℵ
0
exp
−
V
2
− 2αV
√
E
s
cos(Θ
r
− Θ
m
) + α
2
E
s
ℵ
0
p
V,Θ
r
(V, Θ
r
|α)
V p
Θ
r
(Θ
r
|α)
p
Θ
r
(Θ
r
|α) =
∞
0
p
V,Θ
r
(V, Θ
r
|α)V dV
=
1
πℵ
0
∞
0
V exp
−
V
2
− 2αV
√
E
s
cos(Θ
r
− Θ
m
) + α
2
E
s
ℵ
0
dV
α p
Θ
r
(Θ
r
|α)
α
p
α
(α) =
2α
Ω
exp
−α
2
/Ω
, α ≥ 0, Ω = E[α
2
]
Ω = 1 p
ray
Θ
r
(Θ
r
)
p
ray
Θ
r
(Θ
r
) =
∞
0
p
Θ
r
(Θ
r
|α)p(α)dα
=
2
πℵ
0
∞
0
∞
0
αV exp
−
V
2
− 2αV
√
E
s
cos(Θ
r
− Θ
m
) + α
2
(E
s
+ ℵ
0
)
ℵ
0
dV dα
s
m
s
m
[Θ
1
Θ
2
]
P
ray
M
= 1 −
2
πℵ
0
θ
2
θ
1
∞
0
∞
0
αV exp
−
V
2
− 2αV
√
E
s
cos(Θ
r
− Θ
m
) + α
2
(E
s
+ ℵ
0
)
ℵ
0
dV dα dΘ
r
x s
s s
N
p
j
j N
N
U(0, 1) C
i
=
i
j=1
p
j
i
C
i−1
< U(0, 1) ≤ C
i
N
N
F F Q
R C
A = (a
s
k
) m ≥ 2
A =
··· a
0
−1
a
0
0
a
0
1
a
0
2
···
··· a
1
−1
a
1
0
a
1
1
a
1
2
···
··· a
m−1
−1
a
m−1
0
a
m−1
1
a
m−1
2
···
m a
s
k
∈ F F ⊂ C
A
l
A m × m
A
l
= (a
s
lm+r
), r = 0, . . . , m − 1 s = 0, . . . , m − 1
l ∈ Z A
A = (. . . , A
−1
, A
0
, A
1
, A
2
. . .),
A
0
=
a
0
0
··· a
0
m−1
a
m−1
0
··· a
m−1
m−1
A
A(z) =
∞
l=−∞
A
l
z
l
,
A(z)
m × m
A(z) =
k
a
0
mk
z
k
···
k
a
0
mk+m−1
z
k
k
a
s
mk+r
z
k
k
a
m−1
mk
z
k
···
k
a
m−1
mk+m−1
z
k
,
A
A
A(z) =
N
2
l=N
1
A
l
z
l
,
A
N
1
A
N
2
g = N
2
− N
1
+ 1
A
A(z) A(z)
A(z) = A
∗
(z
−l
) =
l
A
∗
l
z
−l
,
A
∗
l
= A
t
A
l
A m
A m
A(z) ·
A(z) = mI,
∞
k=−∞
a
s
k
= mδ
s,0
, 0 ≤ s ≤ m − 1.
δ
s,0
A
A m × mg g
A m ×m
A
m g
W M(m, g; F) F
A
k
a
s
′
k+ml
′
a
s
k+ml
= mδ
s
′
,s
δ
l
′
,l
.
A = (a
s
k
) = (a
s
0
, . . . , a
s
mg−1
)
√
m
m a
0
a
s
1 < s < m
m
m
m m − 1
a
s
k
= 0 0 ≤ k < mg
a
k
:= a
0
k
b
s
k
:= a
s
k
,
0 < s < m 0 ≤ k < mg
a = (a
0
, . . . , a
m−1
)
b
s
= (b
s
0
, . . . , b
s
m−1
)
1 1
1 −1
,
1 1
−1 1
.
1 1
−e
iθ
e
iθ
.
D
2
D
2
=
1 +
√
3 3 +
√
3 3 −
√
3 1 −
√
3
−1 +
√
3 3 −
√
3 −3 −
√
3 1 +
√
3
a
k
(θ)
a
0
(θ) =
1
2
1 +
√
2 cos
θ +
π
4
,
a
1
(θ) =
1
2
1 +
√
2 cos
θ −
π
4
,
a
2
(θ) =
1
2
1 −
√
2 cos
θ +
π
4
,
a
3
(θ) =
1
2
1 −
√
2 cos
θ −
π
4
,
0 ≤ θ < 2π b
k
(θ) = (−1)
k+1
a
3−k
(θ)
A :=
a
0
(θ) a
1
(θ) a
2
(θ) a
3
(θ)
b
0
(θ) b
1
(θ) b
2
(θ) b
3
(θ)
θ,
a
k
b
k
D
2
θ = π/6
H(m; F) = MW (m, 1; F).
H(m; F) m
U(m) m U
m × m U
∗
U = I
H m × m
H =
1 0
0 U
,
U ∈ U(m − 1) m
: =
1 1 ··· ··· ··· ··· 1
−(m − 1)
1
m−1
1
m−1
··· ··· ··· ···
1
m−1
··· ··· ···
0 0 ··· −s
m
s
2
+s
m
s
2
+s
···
m
s
2
+s
··· ··· ···
0 ··· ··· ··· 0 −
m
2
m
2
s = (m − k) k = 0, 1, . . . , m − 1
H = (h
s
r
)
h
r
:= h
0
r
= 1, 0 ≤ r ≤ m.
H
′
H
′′
∈ H(m; C)
U ∈ U(m − 1)
H
′
=
1 0
0 U
H
′′
.
A a
s
k
∈ R A
A =
1 0
0 O
O ∈ O(m − 1) m
A A(z)
χ(A) A
χ(A) := A(1).
A ∈ MW (m, g; F), χ(A) ∈ H(m; F) χ
m m
MW (m, g; F)
χ
→ H(m; F).
H = χ(A)
h
r
s
=
∞
l=−∞
a
r
ml+s
.
H
m−1
s=0
h
r
s
=
m−1
s=0
∞
l=−∞
a
r
ml+s
=
∞
k=−∞
a
r
k
= mδ
r,0
.
H = χ(A)
z = 1
A m χ(A)
A U ∈ U(m − 1)
B =
1 0
0 U
A
χ(B)
χ(B) =
1 0
0 U
χ(A).
A ∈ H(m
′
; F) B ∈ H(m
′′
; F)
A ⊗ B ∈ H(m
′
m
′′
; F).
m
′
= m
′′
= 2
A =
a
0
0
a
0
1
a
1
0
a
1
1
, B =
b
0
0
b
0
1
b
1
0
b
1
1
A ⊗ B =
a
0
0
b
0
0
a
0
0
b
0
1
a
0
1
b
0
0
a
0
1
b
0
1
a
0
0
b
1
0
a
0
0
b
1
1
a
0
1
b
1
0
a
0
1
b
1
1
a
1
0
b
0
0
a
1
0
b
0
1
a
1
1
b
0
0
a
1
1
b
0
1
a
1
0
b
1
0
a
1
0
b
1
1
a
1
1
b
1
0
a
1
1
b
1
1
.
H =
1 1
−1 1
,
H
⊗n
:= H ⊗ ··· ⊗ H
n
,
H
⊗n
2
n
n = 2
H
⊗2
=
1 × 1 1 × 1 1 × 1 1 × 1
1 × (−1) 1 × 1 1 × (−1) 1 × 1
−1 × 1 −1 × 1 1 × 1 1 × 1
−1 × (−1) −1 × 1 1 × (−1) 1 × 1
H
⊗2
=
1 1 1 1
−1 1 −1 1
−1 −1 1 1
1 −1 −1 1
H
⊗2
=
H H
−H H
.
A ∈ MW (m, g; F)
E : MW (m, g) → MW (m, 4g).
a
i
, i = 0, 1, ..., m − 1 A
A =
a
0
a
1
a
m−1
.
m × 4m A
E(A) :=
1
2
a
0
a
1
a
0
−a
1
a
0
a
1
−a
0
a
1
a
m−2
a
m−1
a
m−2
−a
m−1
a
m−2
a
m−1
−a
m−2
a
m−1
.
A ∈ MW (m, g; F) E(A) ∈ MW (m, 4g; F).
E
n m A ∈ W M(m, g; F) E
n
(A) ∈
W M(m, 4
n
g; F)
f : Z → C A
A
f(n) =
m−1
r=0
k∈Z
c
r
k
a
r
n+mk
c
r
k
=
1
m
n
f(n)¯a
r
n+mk
f(n) =
0≤r<m
l
c
r
l
a
r
ml+n
,
n∈Z
|f(n)|
2
= m
0≤r<m
l
|c
r
l
|
2
.
f : Z → C
||f||
2
: =
n∈Z
|f(n)|
2
f(n) =
0≤r<m
l
c
r
l
a
r
ml+n
,
||f||
2
= m
0≤r<m
l
|c
r
l
|
2
.
A ∈ W M(m, g; C) ϕ(x)
m − 1 ψ
1
(x), ..., ψ
m−1
(x) L
2
(R) R
f(x)
R L
2
(R) |f(x)|
2
x∈R
|f(x)|
2
dx < ∞
A
ϕ(x) =
mg−1
k=0
a
0
k
ϕ(mx − k)
a
0
k
A ∈ W M(m, g; C)
A = (a
s
k
) ϕ ∈
L
2
(R)
ϕ
A {ψ
1
(x), ..., ψ
m−1
(x)}
A ϕ
ψ
s
(x) :=
mg−1
k=0
a
s
k
ϕ(mx − k)
A ∈ W M(m, g; C)
ϕ ∈ L
2
(R)
ϕ D.54
R
ϕ(x)dx = 1
ˆϕ ∈ C
0
(R)
sup ϕ ⊂
0, (g − 1)
m
m−1
+ 1
.
ˆϕ ϕ
ϕ
0
:= χ
[0,1)
χ
K
=
1, x ∈ K
0,
ϕ
ν
(x) :=
mg−1
k=0
a
0
k
ϕ
ν−1
(mx − k), ν ≥ 1
ϕ
ν
(x) ϕ(x) ν → ∞
A
A m ϕ ψ
s
s = 1, ..., m −1
k, j ∈ Z
ϕ
jk
(x) := m
j/2
ϕ(m
j
x − k),
ψ
s
jk
(x) := m
j/2
ψ
s
(m
j
x − k), s = 1, ..., m − 1.
W[A] A
W[A] := {ϕ
k
(x), k ∈ Z} ∪ {ψ
s
jk
(x), j, k ∈ Z, j ≥ 0, s = 1, ..., m − 1}.
ϕ
k
(x) := ϕ
0k
(x)
A ∈ W M(m, g; C) W[A]
A f ∈ L
2
(R) L
2
f(x) =
∞
k=−∞
c
k
ϕ
k
(x) +
m−1
s=1
∞
j=0
∞
k=−∞
d
s
jk
ψ
s
jk
(x),
c
k
=
∞
−∞
f(x)ϕ
k
(x)dx,
d
s
jk
=
∞
−∞
f(x)ψ
s
jk
(x)dx.
W[A]
L
2
(R)
W[A]
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