regularity, moments, and wavelet system design

7/29/2019 Regularity, Moments, And Wavelet SYSTEM DESIGN

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N2 1 N

h (n)

h1 (n) (t) (t) (t)


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h (n)

n

h (n) =

2k

h (k) h (k + 2m) = (m) .

K K z = ei

H(z) =

1 + z1

2

KQ (z)

H(z) =

nh (n) zn h (n) Q (z)

z = ei h (n) (t) (t)

K = 1N H(z) N1

z = 1 K Q (z) N1K (t)2

N/2

1 K N2

.

N

N

h (n)H(z)

z = 1 = (t) (t)

kth (t) (t)

m (k) =

tk

(t) dt


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m1 (k) = tk (t) dtkth h (n) h1 (n)

(k) =n

nk h (n)

1 (k) =n

nk h1 (n) .

h (n)

(k, ) = n

(2n + )k h (2n + ) .

(k) = (k, 0) + (k, 1)

m (k) =1

(2k 1) 2k

=1

k

() m (k )

m1 (k) =1

2k

2

k=0

k

1 () m (k ) .

M

(t)K

1 (k) = 0 k = 0, 1, , (K 1)m1 (k) = 0 k = 0, 1, , (K 1)

k = 0, 1, , (K 1)


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= kth = 0

k = 1, 2, , 2K 1(K 1)

(K 1)

(t) K K 1

dk

dtk (t)

< , 0 k K

m1 (k) = 0. 0 k K (t)

L m1 (k) = 0 0 k K 1 dPdtP (t) <

LP > K (t)

h1 (n) (t) H() = 0

(t) (t)

h (n) K =

H() =

1 + ei

2

KL ()

|H() |2 + |H( + ) |2 = 2L () = |L () |2

L () = P

sin2 (/2)


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K N/2

P(y) =

K1k=0

K1 + k

k

yk

+ yK

R1

2 yR (y) P(y) 0 0 y 1R = 0 N K = N/2 N > 2 K

R y

H() = h (n) (t)

h (n) =

H() h (n) |H() |2

|H() |2 =1 + ei2

2K

|L () |2.

M() = |H() |2 L () = |L () |2

M() = |cos2(/2) |K L () .M() L () cos ()

cos () = 1

2 sin2 (/2)

M

sin2 (/2)

= |cos2 (/2) |K Psin2 (/2)y = sin2 (/2) = 1 cos2 (/2)

M(y) = (1 y)K P(y)P(y) (N K)

H()

|H() |2 + |H( + ) |2 = 2

M() + M( + ) = (1 y)K P (y) + yK P(1 y) = 2.


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(N/2 1)K = N/2

P(y) =K1k=0

K 1 + k

k

yk

h (n)

K < N/2 P (y)

P(y) =K1k=0

K 1 + k

k

yk + yK R1

2 y

R (y) NP (y) 0h (n) N

h (n) |H() |2H()

yz cos () = 1 2 y isin () =

cos2 () 1 ei = cos () isin ()

z = ei

H(z) H(1/z) N

1

H(z) N1M

h (n)h (n)

h (n) h (n)

K = N/2 =

K = (N/2) 2

h (n) h1 (n)

N/2 = 4 (0) =

2


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n h (n) h1 (n) (k) 1 (k) k

(t) (t)

N = 6

n h (n) h1 (n) (k) 1 (k) k

N = 4

n h (n) h1 (n) (k) 1 (k) k

N = 2

n h (n) h1 (n) (k) 1 (k) k


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N = 6

k (k) 1 (k) m (k) m1 (k)

N = 4

k (k) 1 (k) m (k) m1 (k)

(0) =

21 (0) = m1 (0) = 0

h (n) h1 (n) m (0) = 1N/2 1 m1 (k)

N/2 1 m1 (0) = 0 N/2m (2) = m(1)

2

N = 4, 6, 8, 10, 12, 16, 20, 40


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h (n)N

N = 2 N = 4 N = 6N = 14 h (n)

h (n)


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N = 4, 6, 8, ...,40


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(t)


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N = 4, 6, 8, ...,40


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|H() |2

N > 2 KR (y)

H() = /2 =

H() = 0

M(0) = 0d M()

d

=0

= 0

h (n)

H()

h (n)

o

o

o : IR < < o

| | | | x, y IR > 1

> 1 or > 0

o : R C r = P +


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r o L2

Hs+1/2 Cr Hs

oo

limN

NN

=

1 log (3)

2 log (2)

= (0.2075 )

o

oo

o


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o +


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= +

=


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N = 26 SH = 4.005SH = 5.06

M = 2 M

H() (t) (t) h (n)

nh (n) = 2 (t) dt = 1 K 2

m (2) = m2 (1) .

f(t) Vj

Pj{f(t)} =k

< f (t) , j,k (t) > j,k (t)

f(t) Vj f(t)Vj

th (t)

m1 () =

t (t) dt.

f(t)

m1 () = 0 = 0, 1,

, L L2

1 = f(t) Pj{f(t)}2 C2j(L+1),C j L f(t)


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f(t)f(t)

Sj{f(t)} =k

2j/2 f

k/2j

j,k (t)

m () =

t (t) dt

m () = 0 = 1, 2, , L L2

2 = Sj{f(t)} Pj{f(t)}2 C2 2j(L+1),

C2 j L f(t)

f(t) j

f(t)

Vjf(t) L2 Vj


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m () = m1 () = 0 = 1, 2, , L m1 (0) = 0 L2

3 = f(t) Sj{f(t)}2 C3 2j(L+1),C3 j L f(t)


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f(t)


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tk (t) dt = m (k) = 0, k = 1, 2, , L 1

tk (t) dt = m1 (k) = 0, k = 1, 2, , L 1.

L 1L 1 m1 (0)

Lh (n) N

N/2 1

h =3 + 7

16

2, 1 7

16

2, 7 7

8

2, 7 + 7

8

2, 5 + 7

16

2, 1 7

16

2

,

h =

3 7

16

2,

1 +

7

16

2,

7 +

7

8

2,

7 78

2,

5 716

2

,1 +

7

16

2

,

h =

3 + 15

16

2,

1 1516

2

,3 15

8

2,

3 +

15

8

2,

13 +

15

16

2,

9 1516

2

,

h =

3 15

16

2,

1 + 1516

2

,3 + 15

8

2,

3 158

2,

13 1516

2

,9 + 15

16

2

,

(k) 1 (k) k = 0, 1, 2, 3

1 (0) = 0

m (2) = m(1)2

N = 6 = 1, 2, 3,

H() h (n)

= /3


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o

K 1h (n) H() =

nh (n) e

in 2K 1 = 0 kth k 1 < k < 2K 1

k

=0

k

(1)

() (k ) = 0.

(k) (0) =

2 k

(k) =1

2

2

k1=1

k

(1) () (k )

k = 2, 4, , 2K 2

(2) = 12

2 (1)

(4) = 122

8 (1) (3) 34 (1)

k=0

k

(1)km () m (k ) = 0

0 < k < 2K K 1 m (k)k

m (k) =12

k1=1

k

(1)m () m (k )


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k = 2, 4, , 2K 2

m (2) = m2 (1)

m (4) = 4 m (3) m (1) 3 m4 (1)m (6) = 6 m (5) m (1) + 10 m2 (3) + 60 m (3) m3 (1) + 45 m6 (1)

m (8) = 8 m (7) m (1) + 56 m (5) m (3) 168 m (5) m3 (1)+2520 m (3) m5 (1) 840 m (3) m2 (1) 1575 m8 (1)


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N = 6 L = 2

n h (n) h1 (n) (k) 1 (k) k

N = 8 L = 3

n h (n) h1 (n) (k) 1 (k) k

N = 12 L = 4

n h (n) h1 (n) (k) 1 (k) k

k = K 1 m (1) = m (3) = 0m (2) = m (4) = m (6) = m (8) = 0


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m (8)

N = 6 L = 2k (k) 1 (k) m (k) m1 (k)

N = 8 L = 3

k (k) 1

(k) m (k) m1

(k)

m (4) = (4) = 0

N = 6 + 2 (t) (t)


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N = 6 + 2

N = 6N

N = 6 2

N = 6 2

N = 4 L = 1

n h (n) h1 (n) (k) 1 (k) k

N = 10 L = 3

n h (n) h1 (n) (k) 1 (k) k

N = 10, 16, 22, 28

N = 6 + 2 N = 6


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N L m = 0 m1 = 0 m = 0 m1 = 0 o

N Lm1 (0) = 0

m1 (0) = 0N = 6 + 2

o N L

NN h (n) (t)

N/2 N/2 1N/2 1

(N/2 1) /2

2N/3N/2


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N = 6 + 2N = 6

N = 6 2

o

N m = 0 m1 = 0

N = 6 + 2 (N 2) /3 (N 2) /3 (2/3) (N 2)N = 6 N/3 (N 3) /3 (2/3) (N 3/2)N = 6

2 (N + 2) /3 (N

4) /3 (2/3) (N

1)

= 1, 2, 1 (0) = 0

N = 6 2

o

N/2

1

o


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regularity, moments, and wavelet system design

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