Global and local spectral analysis of oscillating signals and images

Global'nyj i lokal'nyj spektral'nyj analiz ostsilirujushchich signalov i izobrazhenij


Textbook, 2014
88 Pages

Free online reading

1
2
1.1 . . . . . . . . . . . . .
2
1.2 ;
. . . . . . . . . . . . . . . . . .
9
1.3 - -
. . . . . . . . . . . . . . . . . .
17
1.4 -
. . . . . . . . . . . . . . . . .
24
1.5
. . . . . . . . . . . . . . . . . . . . . . . .
32
2 -
-
38
2.1 . . . . . . . . . . . . . . . . . . .
38
2.2 42
2.3 :
. . . . . . . . . .
45
3
48
3.1 . . . . . . . .
48
3.2
. . . . . . . . . . . . . . . . .
57
1

3.3 . . . . . . . .
63
3.4
-
. . . . . . . . . . . . . . . .
64
3.5 -
71
3.6 - -
: . . . . . .
76
3.7
. . . . . . . . . . . . .
81
2

1
1.1
-
, sin(t) cos(t) 2,
.. sin(t + 2n) = sin(t), cos(t + 2n) = cos(t), n Z
sin
2
(t) + cos
2
(t) = 1, , -
-
.
-
,
exp(it) = cos(t) + i sin(t),
3

, f(t)
f (t) = f (t
0
) +
df
dt
t=t
0
(t
- t
0
) +
d
2
f
dt
2
t=t
0
(t
- t
0
)
2
2
+
+
d
3
f
dt
3
t=t
0
(t
- t
0
)
3
6
+ ...
d
n
f
dt
n
t=t
0
(t
- t
0
)
n
n!
+ ... (1.1)
­ (
), ­ ( ­ ),
t
0
= 0
d
n
dt
n
sin(t)
|
t=0
=
-(-1)
n
, n = 2m - 1;
0, n = 2m,
m N
sin(t) = t
-
x
3
6
+
t
5
120
- ... - (-1)
n
x
2n-1
(2n
- 1)!
, n N. (1.2)
,
d
n
dt
n
cos(t)
|
t=0
=
0, n = 2m
- 1;
(
-1)
n-1
, n = 2m,
m N
cos(t) = 1
-
x
2
2
+
t
4
24
- ... + (-1)
n
x
2n
(2n)!
, n N.
(1.3)
, ..
d
n
dt
n
e
it
t=0
= i
n
, n N.
e
it
= 1 + it
-
x
2
2
- i
x
3
3
+ ... + i(
-1)
n-1
x
n
n!
, n N.
(1.4)
, (1.4) ­ (1.2) (1.3).
4

0
5
10
15
20
-1
0
1
-1
0
1
t
exp(it)
cos(t)
i
sin(t)
0
5
10
15
20
-1
0
1
-1
0
1
t
exp(-it)
cos(t)
-i
sin(t)
. 1.1.
, exp(±it) = cos(t) ± isin(t) ( ).
( ) (- ) ­
- , -
­ ,
.
-
-
­ :
cos(t) =
e
it
+ e
-it
2
,
sin(t) =
e
it
- e
-it
2i
.
(1.5)
(1.5)
, . . 1.1: , -
"-
" () "-
" () . -
.
5

x = sin(t), a y = cos(t).
dx
dt
= y,
dy
dt
=
-x,
, -
.
, -
d
2
x/dt
2
+ x = 0, x -
.
sin(t), cos(t) exp(it) ­
2, sin(nt), cos(nt) exp(int) ­ -
( 2/n). n ­ -
( ), 2 ­
( 2 -
n 2/n). ,
f (t) =
+
n=-
c
n
e
int
(1.6)
2.
6

, () -
c 2 :
f (t) = f
0
+
+
n=1
a
n
sin(nt) + b
n
cos(nt).
(1.7)
, (1.5) -
(1.6), c
0
= f
0
, c
n
=
(b
n
- ia
n
) /2, c
-n
= (b
n
+ ia
n
) /2.
, (1.6), (1.7) -
: c
n
= c
-n
, ..
, -
, a
n
= 0 ­ -
(
); c
n
=
-c
-n
, ,
b
n
= 0, f
0
= 0
.
-
. 1.2, 1.3. , -
-.
(1.6), (1.7) -
. -
-
1
.
(1.6), (1.7)
.
, -
f(t), -
.
1
, (),
() . : Jerri A. The Gibbs Phenomenon in Fourier
Analysis, Splines and Wavelet Approximations. Springer, 1998.
7

-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
f(t)
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
(-2/
)sin(2
t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
n=1
2
(-2/
n)sin(2
nt)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
n=1
5
(-2/
n)sin(2
nt)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
n=1
10
(-2/
n)sin(2
nt)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
t
n=1
100
(-2/
n)sin(2
nt)
. 1.2. -
.
8

-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
f(t)
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+(4/
)cos(2
t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+
n=1
2
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+
n=1
5
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+
n=1
100
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
t
1+
n=1
100
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
. 1.3. - -
.
9

1.2 ;
-
, -
, -
,
-
.
, -
exp(i
0
t) -
exp(-it), ,
, t,
K(, t) =
1,
=
0
;
exp [i(
0
- )t] , =
0
.
(1.8)
f = exp(2t) -
. 1.4.
, K(, t) -
=
0
, ­ -
,
. , (1.8), ­ -
( ) (
), .. ­ -
, , , .. -
, .
, -
, ,
0
,
. , (. . 1.4)
0
= 2m
( . 1.4 m = 1), = 2n, n Z, -
10

0
5
10
0
5
10
15
-1
0
1
/
t
Re[K(
,t)]
0
5
10
0
5
10
15
-1
0
1
/
t
Im[K(
,t)]
. 1.4.
(1.8)
0
= 2. -
( ) .
, T = 2/
0
,
T
0
K(, t)dt =
T
0
e
2i(m-n)
dt
T, n = m;
0. n = m.
(1.9)
, -
-
, ,
f (x) =
+
n=-
c
n
e
2n
T
x
,
,
11

:
c
n
=
1
T
T
0
f (x)e
-
2n
T
x
dx.
,
,
f (x) = f
0
+
n=1
a
n
sin
2n
T
x + b
n
cos
2n
T
x
,
:
f
0
=
1
T
T
0
f (x)dx,
a
n
=
2
T
T
0
f (x) sin
2n
T
x dx, b
n
=
2
T
T
0
f (x) cos
2n
T
x dx.
-
, , -
2
. :
,
dx
dt
= y + µf (x, y, ; µ),
(1.10)
dy
dt
=
-
2
0
x + µg(x, y, ; µ),
(1.11)
2
Andronov A. Les cycles limites de Poicar ´e et la th ´eorie des oscillations auto-entrenues.
Comptes Rendus de l'Acad ´emie des Sciences de Paris. 189 (1929) 559-561.
12

µ ­ .
, -
, ­
. ,
, .. µ ,
.
(1.10)-(1.11) µ = 0, .. -
­ , -
x = R cos(
0
t), y = -
0
R sin(
0
t).
, -
, 2/. -
(1.10)-(1.11) ,
, -
-
2
0
0
[f (R cos(
0
), -
0
R sin(
0
); 0) cos
0
-
- g(R cos(
0
), -
0
R sin(
0
); 0) sin(
0
)] d = 0, (1.12)
R.
, , -
, .
z = x + i
-1
0
y. -
:
z = R cos(
0
t) - i
-1
0
0
R sin(
0
t) = R exp(-i
0
t).
(1.11) i
-1
0
,
dz
dt
=
-i
0
z + µZ(x, y; µ),
(1.13)
Z(x, y; µ) = f (x, y; µ)+i
-1
0
g(x, y; µ).
13

(1.13) exp(i
0
t)
T = 2/
0
:
2
0
0
dz
dt
e
i
0
t
dt = -
0
2
0
0
ze
i
0
t
dt +
2
0
0
Ze
i
0
t
dt.
,
2
0
0
e
i
0
t
dz
dt
dt = ze
i
0
t
2
0
0
-
2
0
0
z
de
i
0
t
dt
dt = -
0
2
0
0
ze
i
0
t
dt,
( ­ , z(0) = z(2/
0
))
.
2
0
0
Ze
i
0
t
dt = 0,
(1.14)
, ,
(1.10)-(1.11)
, -
0
, ..
c -
(1.10)-(1.11).
(1.12) (1.14) -
µ = 0:
Ze
i
0
t
= (f + i
-1
0
g)(cos(
0
t) + i sin(
0
t)) =
f cos(
0
t) - g
-1
0
sin(
0
t) + i f sin(
0
t) + g
-1
0
cos(
0
t) .
14

, -
, -
(1.12)
(1.14)
3
.
. 1.5.
(1.8)
0
= 2. -
,
0
,
, .
, -
f(t)
F () =
1
2
+
-
f (t)e
-it
dt.
3
, -
,
/2, -
.
15

, , -
(1.8) (. c. 1.5)
(
0
- ) =
+
-
K(, t)dt =
=
+
-
e
i(-
0
)t
dt =
, =
0
;
0.
=
0
.
(1.15)
=
0
-
, ­ , -
, ..
-
.
,
f (t) =
+
-
F ()e
it
d
e
i
0
t
=
+
-
(
0
- )e
it
d.
-
(1.15),
-
,
f (
0
) =
+
-
f ()(
0
- )d,
(1.16)
16

-
f() -
, .
(1.16) : (1.15)
, - , -
, , f() , -
0
, -
.
(1.16) f 1, , -
, ,
, - :
+
-
(
0
- )d = 1.
(1.17)
-
( -
) , -
­ ­ -
, .
,
F () =
1
2
+
-
f (t)e
-it
dt, f (t) =
+
-
F ()e
it
d
(1.18)
-
(.. -
). F () -
- -
( -
, .. (1.6)).
17

0
5
10
15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Re[f(t)]
t
-20
0
20
0
1
2
3
4
5
6
7
8
|F(
)|
. 1.6. (
) () -
().
1.3 -
, , -
, -
: f(t) = exp(i2t), f(t) =
exp(i4t), . . 1.6(). -
, . . 1.6(),
. , -
­ , -
, -
-
.
,
18

exp(-it), , , -
K(, t) =
1,2
t
, . . 1.7.
, , -
-
,
, .. t (-,+),
t ­
(t
- ,t + ).
-
ab initio, -
. , c , -
, ,
K(, t, ) ´o (-
) K(, t, )
,
K(, t, + ) =
1
2
[K(, t
- t,) + K(,t + t,)].
t -
t, ,
(master equation) (1.1),
,
K(, t, ) +
K
=
1
2
2K(, t, ) + 2
2
K
t
2
(t)
2
2
,
(
- , ±t).
, (t)
2
/t =
2
= const -
K
=
2
2
2
K
t
2
,
(1.19)
19

0
5
10
0
5
10
15
-1
0
1
/
t
Re[K(
,t)]
0
5
10
0
5
10
15
-1
0
1
/
t
Im[K(
,t)]
. 1.7. :
. 1.6 exp(-it). : -
, =
1,2
.
20

.
(1.18). -
K(, t, ) =
+
-
^
K(, , )e
it
d,
(1.20)
.
2
K
t
2
=
+
-
^
K(, , )
2
t
2
e
it
d =
+
-
-
2
^
K(, , ) e
it
d.
, -
(1.19) -
^
K
=
-
2
2
2
^
K
^
K = ^
K
0
exp
-
2
2
/2 ,
(1.21)
^
K
0
­ ­ - -
:
^
K
0
(, ) =
1
2
+
-
K(, t
, 0)e
-it
dt
.
(1.22)
(1.22), (1.21) (1.20),
K(, t, ) =
1
2
+
-
+
-
K(, t
, 0)e
-
2
2
/2-i(t
-t)
ddt
.
21

,
,
K(, t, ) =
1
2
+
-
K(, t
, 0)e
-
(t
-t)2
22
dt
+
-
e
-
2
+i
(t
-t)
2
2
d,
4
+
-
e
-
2
d = ,
K(, t, ) =
+
-
K(, t
, 0)e
-
(t
-t)2
22
dt
2
2
.
(1.23)
(1.23) -
, , K, , K(, t
, 0)
t
,
(t-3,t+3) (.
. 1.8).
4
: ,
+
-
e
-
2
d
2
=
+
-
e
-
2
d
+
-
e
-
2
d
=
+
-
e
-
(
2
+
2
)
dd,
,
2
+
2
= r
2
( )
dd = d(r
2
), ( )
( 0 , )
+
0
e
-
r
2
d(r
2
) = - e
-
r
2
0
= -(0 - 1) = ,
.
22

0
t
exp[-t
2
/2
2
]/(2
2
)
1/2
-
1/2
1/2
-2
1/2
-3
1/2
2
1/2
3
1/2
. 1.8. .
(1.23) (1.8). -
, K(, t
, 0) -
(1.23)
( , K
, -
).
+
-
e
i(
0
-)t
e
-
(t
-t)2
22
dt
2
2
=
= e
i(
0
-)t
+
-
e
i(
0
-)(t
-t)-
(t
-t)2
22
dt
2
2
,
-
.
23

,
+
-
e
i(
0
-)t
e
-
(t
-t)2
22
dt
2
2
= e
i(
0
-)t
e
-2
2
(
0
-)
2
,
(1.24)
(1.9) "-
" / -
0
. -
(1.23), , -
, , .
, . ,
-
2
,
.
= 1.
, , -
, -
-
. , . 1.9 -
, -
. , -
,
, -
(1.24)
,
, .
24

0
5
10
15
-1
0
1
-1
0
1
t
Re[f(t)]
Im
[f
(t
)]
t
|K( ,t,1)|
0
5
10
15
2
4
6
8
10
12
14
0.2
0.4
0.6
0.8
. 1.9. ( -
) (. . 1.6) , -
-
(1.23).
1.4 -
-
(1.24) , -
exp(
0
t) , -
0
= . , (1.24)
. (1.24)
exp(it),
e
it
K(, t, ) = e
i
0
t
e
-2
2
(
0
-)
2
-
.
25

0
5
10
15
-1
0
1
t
f(t)
=0.25
0
5
10
15
5
10
15
=1
0
5
10
15
5
10
15
t
=2
0
5
10
15
5
10
15
. 1.10.
( ­ ) , -
exp(i2t) t 5, 0.5exp(i2t) + 0.5exp(i4t) 5 < t 10
exp(i4t) t > 10 ( ­ -
, ­ ). ,
, .
f(t), -
G(, t) =
+
-
f (t
)e
i(t
-t)
e
-
(t
-t)2
22
dt
2
2
,
(1.25)
(
).
,
, -
( ) ( -
, ) -
, . . 1.10.
26

0
2
4
6
8
10
12
14
16
18
0
0.5
1
G(
,2.5)
0
2
4
6
8
10
12
14
16
18
0
0.5
1
G(
,7.5)
0
2
4
6
8
10
12
14
16
18
0
0.5
1
G(
,12.5)
. 1.11. ,
. 1.10, (
G) -
.
, ,
,
,
, -
, . -
, , . 1.10,
. 1.11 -
(1.25) = 1 ( -
­
. 1.10). ,
2 = 2.
27

(
-
) -
.
, -
,
, (1.13), ­
-
(
-
)
dz
dt
= i
0
z + µ 1 - |z|
2
z,
(1.26)
(1.26) z = Z(t) exp(i
0
t+ i
0
),
dz
dt
=
dZ
dt
e
i
0
t+i
0
+ i
0
Ze
i
0
t+i
0
=
dZ
dt
e
i
0
t+i
0
+ i
0
z.
(1.26) -
,
dZ
dt
= µ 1
- Z
2
Z,
(1.27)
Z(0) = Z
0
Z(t) = Z
0
e
µt
Z
2
0
(e
2µt
- 1) + 1
.
(1.28)
: µ ,
,
5
.
5
-
- : Garcia-Morales V., Krischer K. The
complex Ginzburg-Landau equation: an introduction. Contemporary Physics. 53 (2012)
79-95.
28

0
20
40
60
80
100
-1
0
1
-1.5
-1
-0.5
0
0.5
1
1.5
t
Re(z)
Im(z)
. 1.12. ­ -
(1.26)
0
= 1, µ = 0.1, = 1 z = Z
0
= 0.01 ( ),
(1.28) ( ,
- ).
. 1.12.
, . . 1.13. , -
(
),
-
,
- (. (1.28),
Z = 1/ t; , , dZ/dt = 0
, (1.27)).
-
(1.28), -
, ,
, .
29

10
20
30
40
50
60
70
80
90
100
0.5
1
1.5
2
t
|G(
,t)|
10
20
30
40
50
60
70
80
90
100
0
0.5
1
t
|G(1,t)|
10
20
30
40
50
60
70
80
90
100
-2
-1
0
1
2
t
Re[G(1,t)]
10
20
30
40
50
60
70
80
90
100
-2
-1
0
1
2
t
-Im[G(1,t)]
. 1.13. ­
- (. . 1.12): , -
, ( -
) .
- ().
30

,
, .
-
, -
. -
,
,
6
,
dx
dt
= f (x, t, )
-
( ) -
X , -
, -
T :
dX
dt
= ¯
f (X, t, ),
¯
f =
1
T
T
0
f (x, t
, )dt
.
, -
-.
,
-
(, , -
, ­ )
. -
(, , ).
6
. , Jan A.
Sanders (2006) Averaging. Scholarpedia, 1(11):1760, -
: ..
-
( ).
31

-
-
, -
,
-
, -
.
, -
-
, -
, a
,
:
w(t, a, b) =
1
2a
2
e
-
(t-b)
2
2a2
,
´ b
.
, exp [i(t - b)]
. -
, -
, =
0
/a,
0
­ -
,
-
.
(1.25) -
w(a, b) =
+
-
f (t)e
-i
0
(t-b)
a
e
-
(t-b)
2
2a2
dt
2a
2
,
(1.29)
- -
.
,
, ( -
) , . 1.14.
32

-5
0
5
-1
-0.5
0
0.5
1
-5
0
5
-1
-0.5
0
0.5
1
Gabor
-5
0
5
-1
-0.5
0
0.5
1
-5
0
5
-1
-0.5
0
0.5
1
-5
0
5
-1
-0.5
0
0.5
1
Morlet
-5
0
5
-1
-0.5
0
0.5
1
. 1.14. ( ) -
( ) : = 0.5, , 2
( ). , -
0
= (
,
a =
0
/). , -
­ .
1.5 -
,
7
(Morlet wavelet)
() = Ce
i
0
e
-
2
2
,
(1.30)
C ­ ( -
).
7
,
, -
, (1.30) , -
: = C
0
[exp(i
0
) - exp(-
2
0
/2)] exp(-
2
/2)
33

, , -
­
,
+
-
||d = 1.
(1.30) -
,
+
-
||
2
d = C
+
-
e
-
2
2
d = C2 = 1,
C = 1/2. -
(1.29)
= (t - b)/a, d = dt/a, (1.29) -
a
2
.
--
f(t)
.
, , (1.29)
w(a, b) =
+
-
f (t)
()d,
c = (t - b)/a
() = exp[
-i
0
-
2
/2]/2,
- (1.30) .
:
-
2
2
- i
0
= -
1
2
2
+ 2i
0
+ (i
0
)
2
- (i
0
)
2
=
=
-
1
2
( + i
0
)
2
-
2
0
2
.
34

() =
1
2e
-
2
0
2
e
-
1
2
(+i
0
)
2
.
, -
:
w(a, b) = e
-
2
0
2
+
-
f (t)
e
-
[(t-b)+i0a]
2
2a2
2a
2
dt.
(1.31)
(1.31) -
a = 0. ,
8
, -
. -
exp[-
2
/2a
2
]/2a
2
.
,
+
-
e
-
2
2a2
d
2a
2
= 1;
= 0 a 0
, a , = 0
( -
), ­ ,
. :
() = lim
a0
e
-
2
2a2
2a
2
=
, = 0;
0.
= 0,
- .
8
. §V.7 : .., .. -
. .: , 1973.
35

, (1.16)
lim
a0
+
-
f (t)
e
-
((t-b)+i0a)
2
2a2
2a
2
dt =
+
-
f (t)(t - b)dt = f(b).
,
- a = 0
w(0, b) = f (b)e
-
2
0
2
,
(1.32)
-
, -
.
(1.32) ,
f (b) 1, -
+
-
()d =
+
-
()d = e
-
2
0
2
,
(0.0072
0
= , 2.7
· 10
-9
0
= 2), -
9
.
-
.
9
- -
, -
a
b (. §2.4 : . . : "
", 2001.). , ,
(. . 32) ( -
). , ,
"-" -
,
-.
36

,
f (t) = exp(it), -
(1.29) , -
-
:
w(a, b) =
+
-
e
it
e
-i
0
(t-b)
a
e
-
(t-b)
2
2a2
dt
2a
2
=
= e
ib
+
-
e
i
(
-
0
a
)
(t-b)
e
-
(t-b)
2
2a2
dt
2a
2
=
= e
ib
e
-
(a-0)
2
2
+
-
e
-
[(t-b)+ia(a-0)]
2
2a2
dt
2a
2
=
= e
ib
e
-
(a-0)
2
2
. (1.33)
, -
´ ( b) -
, -
( a). -
a
max
, -
a
max
-
0
= 0. , -
-
T = 2/T , ,
-
:
a
max
=
0
2
T.
(1.34)
(1.34) ,
­ ,
37

. 1.15. -
, ( -
. . 1.10) ,
0
=
( )
0
= 2 ( ). -
.
, ­ .
0
= 2 -
.
. 1.15
,
1 0.5 -
. , , -
-
( ).
, , , ´ -
,
..
.
38

2
-
2.1
- -
-
, -
, -
-
39

-
, -
1
.
,
,
(, , -
)
2
.
-
() = exp(
-i
0
-
2
/2)
(- , -
(1.29);
):
() = e
i
0
-
2
2
(
-i
0
- ) = -
()(i
0
+ ),
() =
-
()(i
0
+ )
-
().
,
-, a and b,
, = (t - b)/a,
a
=
a
=
t - b
-a
2
=
-
a
,
b
=
b
=
-
1
a
,
1
: .. -
- -
. . . . . . 2006 46 77-78
( , : ..
"" 2005
128 752-759.)
2
: Haase M.A. Family of Complex Wavelets
for the Characterization of Singularities. In: Paradigms of Complexity, ed. M.M Novak.
World Scientific, 2000, pp.287-288. , -
-,
40

2
b
2
=
-
1
a
b
=
1
a
2
=
-
1
a
2
()(i
0
+ ) +
() . (2.1)
, -
. -
, , -
a
-1
,
w
a
=
a
1
a
+
-
f (t)
()
dt
2=
-
1
a
2
+
-
f (t)
()
dt
2 +
1
a
+
-
f (t)
()
-
a
dt
2.
(2.1) -
,
2
w
b
2
=
-
1
a
3
+
-
f (t)
()(i
0
+ )
dt
2 +
+
-
f (t)
()
dt
2,
a ,
a
2
w
b
2
=
-
i
0
a
2
+
-
f (t)
()
dt
2
i
0
w
b
-
1
a
2
+
-
f (t)
()dt +
+
-
f (t)
()
dt
2
w
a
.
41

,
:
w
a
= a
2
w
b
2
- i
0
w
b
.
(2.2)
(2.2) -
, ´y
, b, "
", a ­ " -
". (2.2) -
( ,
),
exp(
-i
0
) .
, (2.2) -
, ­
(1.32). , -
" ", -
(2.2) -
w(0, b) = f(b), -
exp(
-
2
0
/2)
3
- -
a b. -
(2.2).
3
(1.32), , -
, , (
0
= 2) . (2.2) -
, (1.32),
´ .
42

2.2 -
a = 0 (2.2) -
w
a
=
-i
0
w
b
,
(2.3)
, ,
- .
,
w(a, b) = u(a, b) + iv(a, b) (2.3) -
:
u
(a/
0
)
=
v
b
,
v
(a/
0
)
=
-i
u
b
.
,
(1.33) -
:
a
e
ib
e
-
(a-0)
2
2
=
-(a -
0
) w;
b
e
ib
e
-
(a-0)
2
2
= iw,
a 0 (2.3).
, u = cos(t) u = cos(t) -
exp(it),
f = exp(it) (2.2) , f = exp(iz),
z = b + ia/
0
­ ,
- (2.3), (2.2) -
.
43

, (-
) , , -
u(t), , -
, -
, a = 0 (2.3).
­ , -
­ ,
, , , -
­ , ..
. -
(1.5),
cos(t) =
e
it
2
+
e
-it
2
,
sin(t) =
-i
e
it
2
+ i
e
-it
2
,
, -
() -i, -
c (-) i,
, -
.
, - ^u()
^v() -
^
v() = -isgn()^u(),
sgn() =
1,
0;
-1, 0.
44

, -
,
: -
-
- :
^
F
+
-
f (t
)g(t
- t
)dt
= 2 ^
f ()^
g().
(2.4)
, , -
sgn():
1
2
+
-
h(t)e
-it
dt = -isgn().
, -
, § 1.3
, ,
-
, ,
-it
1
2
+
-
h(t)e
-it
dt = -i2(),
:
t
+
-
1
2
+
-
h(t)e
-it
dt
e
it
d = 2
+
-
()e
it
d,
h(t) = 2/t , , -
,
:
v(t) = H [u(t)] =
1
+
-
u(t
)
1
t
- t
dt
.
(2.5)
45

, - -
f(t) -
(2.2) , -
a = 0, .. (2.2)
c -
: w(0, b) = f(t) + H [f(t)].
-
A(b) =
f (b)
2
+ (H [f (b)])
2
,
(b) = arctg
H [f (b)]
f (b)
w(0, b) =
A(b) exp(i(b)) - -
,
4
2.3 :
(2.2),
w
a
= a
2
w
b
2
- i
0
w
b
,
, , , ­
, -
. -
­ ,
4
-
,
( ) -
: Vela-Arevalo, L.V. (2002) Time-frequency analysis based on wavelets
for Hamiltonian systems. Dissertation (Ph.D.), California Institute of Technology.
http://resolver.caltech.edu/CaltechETD:etd-03302004-115559
46

w ,
. -
, -
, -
. , , -
,
0
, ,
-
. -
w
a
= a
2
w - i
0
· w,
(2.6)
, ,
0
= (
0x
,
0y
).
( ) (2.6)
w
a
= a
2
W
b
2
x
+
2
W
b
2
y
- i
0x
W
b
x
+
0y
W
b
y
.
(2.7)
b
x
, b
y
­ ,
- b = (b
x
, b
y
),
, - a ­ .
(2.7)
w(a, b
x
, b
y
) =
+
+
f (x, y)
(
x
,
y
)dxdy,
(
x
,
y
) =
1
2a
2
e
i(
0x
x
+
0y
y
)
e
-
1
2
(
2
x
+
2
y
)
.
(2.8)
, (2.7) (1.32):
W (0, b
x
, b
y
) =) = f (b
x
, b
y
)e
-
1
2
(
2
0x
+
2
0y
)
= f (b
x
, b
y
)e
-
2
0
2
,
47

0x
=
0
cos(),
0y
=
0
sin(), -
( -
, -
).
,
§ 2.1
5
, , (2.8)
(1.30)
.
5
. : Postnikov E.B., Singh V.K. Local spectral
analysis of images via the wavelet transform based on partial differential equations.
Multidim. Syst. Sign. Proces, 2012 (doi:10.1007/s11045-012-0196-1)
48

3
3.1
, -
MATLAB
, -
(Fast Fourier Transform, FFT), -
fft .
F () =
1
2
+
-
f (t)e
-it
dt
F
n
=
N
j=1
f
j
e
-
2i(j-1)(n-1)
N
,
(3.1)
f
j
­ ,
. , f(t)
49

[t
min
t
max
, f
j
= f (t
j
) t
j
, -
N - 1 (t
1
= t
min
, t
N
= t
max
).
fft -
(Cooley-Tukey), N -
(.., , 2
6
= 64, 2
10
= 1024
..). N (.. n = 1..N), -
.
. 3.1 -
( t
max
N):
N=64;
tmax =5;
t=l i n s p a c e ( 0 , tmax ,N) ;
nu =1;
f =exp ( i 2pinu t ) ;
F= f f t ( f ) ;
, . 3.1 (3.1),
: fft ,
, , -
, , ,
, ­
, , -
, ;
2(n - 1)/N, n = 1..N
(-
­ - -
), -
, "", .. -
, -
/ ­ -
, ­ -
50

0
5
10
-1
0
1
t
f(t)
0
50
100
0
50
100
150
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
50
100
0
50
100
150
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
50
100
0
50
100
150
n
|F
n
|
. 3.1. ­ ( -
-
, ; -
), .
, (.
)
-
, -
.
,
,
, .
, , -
.
, , ,
, ; ,
.
51

0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
. 3.2. , ( ) -
fft : ,
.
( ,
,
­ -
(Nyquist));
+
, -
,
-
=
-
-
, (3.1)
fft , -
(, ,
(1.5)), ,
n (, , F
n
) -
.
-
, . 3.2.
52

- (
(1.5))
F [cos(
0
t)] = 0.5( -
0
) + 0.5( +
0
)
F [sin(
0
t)] = -0.5i( -
0
) + 0.5i( +
0
)
.
- -
,
­ , -
, ­
. , . 3.2, fft
, (
, ),
­ ­ -
. , ,
( -
,
0 2), -
-
.
-
MATLAB
fftshift , . . 3.2
:
N=256;
tmax =20;
t=l i n s p a c e ( 0 , tmax ,N) ;
nu =1;
f =exp ( i 2pinu t ) ;
F0= f f t ( f ) ;
F= f f t s h i f t ( F0 ) ;
53

0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
. 3.3. , ( . 3.2)
, fftshift .
, - -
, -
, ­ -
, ­ .
-
.
, , , -
-N/2 N/2-1
( ,
N/2 -
-N/2)
max
= t
max
- t
min
, 2/T
max
-
-
54

.
, -
. -
+
-
|f(t)|
2
dt =
1
2
+
-
|f()|
2
d
(3.2)
-
´ .
, ,
,
( -
MATLAB
-
trapz).
,
, -
-
, ­
, .
function [ omega , F]= F o u r i e r ( t , f )
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
Fk= f f t ( f i ) ;
Fks= f f t s h i f t ( Fk ) ;
omega=([-N/2:N/2-1])2 pi /( t i (end)- t i ( 1 ) ) ;
fNorm=trapz ( t i , abs ( f i ) . ^ 2 ) ;
FNorm=trapz ( omega , abs ( Fks ) . ^ 2 ) ;
F=Fkssqrt (2 pifNorm/FNorm) ;
55

, -
.
,
MATLAB
-
-, -
, tv (:,1)= t ;
fv (:,1)= f ;
­ .
-
-
( fft
),
( fi =interp1q(tv,fv, ti ); )
, -
( linspace) -
- ( -
N=2^(ceil(log2(n)));, ceil ­ -
´ ).
Fourier , -
, .3.4. , -
, -
-
. ,
( -
),
( ). -
, ­
;
-
­ .. - ).
56

0
5
10
-1
0
1
t
f(t)
-5
0
5
0
5
/2
|F(
)|
-2
0
2
0
5
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-10
0
10
0
5
/2
|F(
)|
-2
0
2
0
5
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-5
0
5
0
5
10
/2
|F(
)|
-2
0
2
0
5
10
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-2
0
2
-10
0
10
/2
Re[F(
)]
-2
0
2
-10
0
10
/2
Im[F(
)]
-2
0
2
-10
0
10
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-2
0
2
-10
0
10
/2
Re[F(
)]
-2
0
2
-10
0
10
/2
Im[F(
)]
-2
0
2
-10
0
10
/2
|F(
)|
. 3.4. , . (3.1)
(3.2) Fourier.
±2, -
, 2.
57

. 3.5. -
.
3.2
, , -
,
, -
, ( -
) . , -
,
f (x, y) = sin(10x) + sin(5y) + sin(7(x + y)),
(3.3)
. 3.5 .
58

F (
x
,
y
) =
1
4
2
+
-
+
-
f (x, y)e
-i
(
x
x+
y
y
)dxdy.
(3.4)
-
MATLAB
fft2 , -
, , fft , , -
-
.
function [ omegax , omegay , F2]= F o u r i e r 2 ( x , y , f )
[ X, Y]= meshgrid ( x , y ) ;
nx=length ( x ) ;
ny=length ( y ) ;
Nx=2^( c e i l ( log2 ( nx ) ) ) ;
Ny=2^( c e i l ( log2 ( ny ) ) ) ;
x i ( : , 1 ) = l i n s p a c e ( x ( 1 ) , x ( end ) , Nx ) ;
y i ( : , 1 ) = l i n s p a c e ( y ( 1 ) , y ( end ) , Ny ) ;
[ Xi , Y i ]= meshgrid ( x i , y i ) ;
f i =interp2 (X, Y , f , Xi , Yi , ' l i n e a r ' , 0 ) ;
F=f f t 2 ( f i ) ;
F2= f f t s h i f t (F ) ;
omegax=([-Nx/2:Nx/2-1])2 pi /( x (end)-x ( 1 ) ) ;
omegay=([-Ny/2:Ny/2-1])2 pi /( y (end)-y ( 1 ) ) ;
-
x y ,
,
- -, .
- -
-
59

. 3.6.
( ,
­ ). , -
x
= 0,
y
= 0
x
=
y
, ..
.
, , , -
.
, Fourier2
. , -
,
, trapz
­ ­ -
. ,
-
, -
.
(3.3)
. 3.6.
60

- 6 -
.
-
, . 3.6. -
, -
, -
­ (. (3.3))
, , .
-
, . 3.6
-
, ( ,
-
, .. , -
- ). -
, -
, -
-
, . -
.
[ ox , oy , F]= F o u r i e r 2 ( x , y , f ) ;
[OX,OY]= meshgrid ( ox , oy ) ;
rho_max=sqrt ( ox ( end)^2+ oy ( end ) ^ 2 ) ;
rho=l i n s p a c e ( 0 , rho_max , 1 2 8 ) ;
p h i=l i n s p a c e ( 0 , 2 pi ,101);
[RHO, PHI]= meshgrid ( rho , p h i ) ;
OXR=RHO. cos (PHI ) ;
OYR=RHO. sin (PHI ) ;
absF=abs (F ) ;
pF=interp2 (OX, OY, absF ,OXR,OYR, ' n e a r e s t ' , 0 ) ;
61

0
1
2
0
5
10
15
0.5
1
30
210
60
240
90
270
120
300
150
330
180
0
. 3.7. - (. . 3.6) -
,
.
(
x
,
y
)
(, ),
=
2
x
+
2
y
, = arctg
y
x
;
x
= cos(),
y
= sin(),
0 2 (.. 360
0
, . . 3.7 ).
-
( 128 × 101), -
interp2 (
' nearest ' -
; , -
: ' linear ', 'cubic', ' spline ').
. 3.7 .
,
62

(
)
, ().
,
( = 0) c
0
= 10, -
( = /2 = 90
0
) ­
/2
= 5
( = /4 = 45
0
) ­
/2
= 72
9.9 ( -
, , ).
-
, . . 3.7 . -
-, -
, |F(,)|
2
, ..
|F|
2
0
|F(,)|
2
d.
. 3.7
IpF=trapz ( pF . ^ 2 ' ) ;
polar ( phi , IpF /max( IpF ) ) ;
, -
, trapz -
-
; ,
, -
-
.
. 3.7 -
, -
(3.3), .
, , ,
63

-
,
, .
-
(3.2), -
,
. , -
(, -
) .
, .
3.3
MATLAB
ifft ifft2 -
.
-
, .. ( -
) -, .
F=fft(f ) f=fft2(F),
f= ifft (F) f=ifft2 (F) -
.
, -
F , -
(.. -
). -
, ,
(2.4),
.
64

3.4 -
-
-
(1.25),
G(, t) =
+
-
f (t
)e
i(t
-t)
e
-
(t
-t)2
22
dt
2
2
,
, -
, -
(2.4),
( -
^
F
-1
) (-
^
F )
( 2):
G(, t) = ^
F
-1
^
F [f (t)] ^
F e
-it
e
t2
22
2
2
2
.
^
f (
) = ^
F [f (t)] -
, § 1.3, -
:
G(, t) = 2 ^
F
-1
^
f (
)e
-2
2
(
-)
2
.
(3.5)
, -
-
, -
MATLAB
-
, .
65

F= f f t ( f ) ;
nrm=2pi /( t (end)-t ( 1 ) ) ;
omega_ = ( [ ( 0 : (N/ 2 ) ) ((( -N/2)+1): -1)])nrm ;
for k=1: length ( omega ) ;
window=exp(-2sigma ^2(omega_+omega(k ) ) . ^ 2 ) ;
c o n v o l ( : , k)=window . F;
G( : , k)= i f f t ( c o n v o l ( : , k ) ) ;
end
,
ifft 1/2,
^
F
-1
, 2 ^
F
-1
. ,
, omega_, -
­
(
§ 3.3) -
fft .
t [0, 1] -
-
f (t) = e
-4t
cos(20t) + [
1
3
,
2
3
](t) sin(40t),
(3.6)
[t
b
,t
e
]
(t) =
1, t
[t
b
, t
e
];
0, t /
[t
b
, t
e
].
(3.6), . 3.8 , -
-
: -
(
)
(
).
66

. 3.8. -
= 0.05,
. ­ -
, .
-
,
, . 3.8 .
,
, , .
,
­ . -
,
fft
, ,
20, , -
,
.
-
67

-0.5
0
0.5
1
1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
t
f(t)
. 3.9. -
.
­ ,
-
, . . 3.9.
, fft ,
,
, .
-
, -
.
fftGabor, , Fourier, -
, -
,
. -
. 3.10. .
68

. 3.10. ,
.
function [ t i ,G]= f f t G a b o r ( t , f , omega , sigma ) ;
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
f p ( 1 :N/2)= f l i p u d ( f i ( 2 :N/ 2 + 1 ) ) ;
f p (N/ 2 + 1 : 1 . 5 N)= f i ;
f p ( 1 . 5 N+1:2N)=flipud ( f i (N/2:N-1));
F= f f t ( f p ) ; nrm=pi / ( t ( end)-t ( 1 ) ) ;N=2N;
omega_ = ( [ ( 0 : (N/ 2 ) ) ((( -N/2)+1): -1)])nrm ;
for k=1: length ( omega ) ;
window=exp(-2sigma ^2(omega_+omega(k ) ) . ^ 2 ) ;
cnv ( k , : ) = window . F;G(k ,:)= i f f t ( cnv (k , : ) ) ;
end
G=G( : ,N/4+1:3N/4);
69

- -
, ,
:
function [ t i ,G]= f f t M o r l e t ( t , f , a , omega0 ) ;
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
f p ( 1 :N/2)= f l i p u d ( f i ( 2 :N/ 2 + 1 ) ) ;
f p (N/ 2 + 1 : 1 . 5 N)= f i ;
f p ( 1 . 5 N+1:2N)=flipud ( f i (N/2:N-1));
F= f f t ( f p ) ;N=2N;
nrm=pi / ( t ( end)-t (1))
omega_ = ( [ ( 0 : (N/ 2 ) ) ((( -N/2)+1): -1)])nrm ;
i f a (1)==0
w( 1 , : ) = f p exp(-omega0 ^2/2);
k1 =2;
else
k1 =1;
end
for k=k1 : length ( a ) ;
omega_s=a ( k ) omega_ ;
window=exp( -(omega_s-omega0 ) . ^ 2 / 2 ) ;
cnv ( k , : ) = window . F;
w( k , : ) = i f f t ( cnv ( k , : ) ) ;
end
w=w ( : ,N/4+1:3N/4);
­ -
,
a = 0 -
, (1.32).
70

. 3.11. -
0
= 2: , -
(
).
71

3.5 - -
-
-
-
(2.2),
, u(a, b)
v(a, b) -
w(a, b) = u(a, b) + iv(a, b) (2.2) -
a
u
v
=
b
a
u
b
+
0
v
a
v
b
-
0
u
,
(3.7)
-
MATLAB
pdepe.
, ,
, -
. , -
-
. ,
. -
, -
. , -
,
;
. -
,
-
, , -
72

,
, -
..
,
-
, , -
-
.
, -
-
/ , -
. -
,
a = 0, :
u(a, ·)
v(a, ·)
+
a
10
-n
+ a
n
b
1
0
0 b
2
a
u
b
+
0
v
a
v
b
-
0
u
=
u(0, ·)
v(0, ·)
.
n ­ -
4 8,
[0, 1] (
). b
1
b
2
{b
1
= 1, b
2
= 0
} {b
1
= 0, b
2
= 1
}, -
.
(3.7) (3.5)
, , -
MATLAB
' -
pdepe, -
(..
,
),
.
73

- :
function w=pdeMorlet ( b , u , omega0 , a , lB , rB , n )
global xu0 uu0 xmax omega u l e f t
global u r i g h t LeftB RightB pow
omega=omega0 ; pow=n ; bmax=length ( b ) ;
xu0=b/b ( bmax ) ; au=a /b ( bmax ) ;
uu0 ( : , 1 ) = r e a l ( u ) ; uu0 ( : , 2 ) = imag ( u ) ;
LeftB=lB ; RightB=rB ;
u l e f t =uu0 ( 1 , : ) ; u r i g h t=uu0 ( bmax , : ) ;
m = 0 ;
s o l = pdepe (m, @pde , @ic , @bc , xu0 , au ) ;
w=( s o l ( : , : , 1 ) + i sol ( : , : , 2 ) ) ;
w=wexp(-omega0 ^2/2);
function [ c , f , s ]= pde ( x , t , u , DuDx)
global omega
c = [ 1 ; 1 ] ;
f = [ t DuDx(1)+omegau (2 );
t DuDx(2)-omegau (1)] ;
s = [ 0 ; 0 ] ;
function u0=i c ( x )
global xu0 uu0
u0 =[ interp1q ( xu0 , uu0 ( : , 1 ) , x ) ;
interp1q ( xu0 , uu0 ( : , 2 ) , x ) ] ;
function [ pl , ql , pr , qr ]= bc ( xl , ul , xr , ur , t )
global u l e f t u r i g h t LeftB RightB pow
e p s l n =10^(-pow ) ;
p l = [ u l (1) - u l e f t (1 ) ; ul (2)- u l e f t ( 2 ) ] ;
q l = [ LeftB ( 1 ) ; LeftB ( 2 ) ] t /( t ^pow+epsln ) ;
p r = [ ur (1) - uright (1 ); ur(2)- uright ( 2 ) ] ;
qr = [ RightB ( 1 ) ; RightB ( 2 ) ] t /( t ^pow+epsln ) ;
74

, , -
b, u,
­ \omega_0
a ( , -
,
­ -
),
lB rB, -
b
1
, b
2
[b1, b2], n
(3.5) .
, -
(3.7) (pde), (ic) (bc) -
. (pdeMorlet)
-
(global), -
. ­ (-
) -
w=(sol (:,:,1)+ isol (:,:,2)); , -
(w=wexp(-omega0^2/2);) -
( -
-
, exp(-
2
0
/2).
-, -
b a -
.
" ", .. ,
, , -
, -
, ( -
, pdepe, -
-
75

0
0.2
0.4
0.6
0.8
1
-1
0
1
t
f(t)
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
a
0
0.01
0.02
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
a
0.1
0.2
0.3
. 3.12. ( ) ,
0.3 cos(20t) sin(40t) ­ , -
-, -
() ().
1
).
, , -
, , -
, ,
(
0
= ). -
. . 3.12, , -
- -
-
.
1
: .., ..
- , -
.
XIX . . 2. - .: , 2007, C. 7-9;
http://spravka.akin.ru/Docs/Rao/Ses19/AI2.PDF
76

3.6 - -
: -
-
, -
.
, , , -
, -
-
. ­ -
§ 2.3, ­ , -
-
.
, , -
w(b
x
, b
y
, a, ) ­ -
,
(b
x
, b
y
), a . , -
.
, - -
( ) -
. -
w(b
x
, b
y
, a, ), -
,
a
-
w
a,
(b
x
, b
y
).
,
. 3.13, -
( -
77

. 3.13. ,
:
,
-, ­ -
.
sin(10x), sin(5y) sin(2(x + y))),
.
.
, -
-
(. . 3.13). -
, -
- -
( = 0, /4, /2), -
­ , -
( [mx,imx]=max(pF');
­ -
78

, ,
plot(phi/pi,rho(imx)/pi) -
; § 3.1). ,
,
.
,
- -
(??),
w(b
x
, b
y
, a, ) =
=
+
-
f (x, y)e
-i
0
a
[(x-b
x
) cos +(y-b
y
) sin ]
e
-
(x-bx)
2+(y-by)2
2a2
dxdy
2a
2
,
=
0
/a .
,
w
,
(b
x
, b
y
) = e
i(b
x
cos +b
y
sin )
×
×
+
-
f (x, y)e
-i(xcos+y sin)
e
-
(x-bx)
2+(y-by)2
2(0/)2
dxdy
2a
2
. (3.8)
(3.8) , -
§§ 1.4-1.5: -
, .
-
, -
, -
.
,
, , ,
­ -
.
79

MATLAB
fspecial imfilter .
. 'gaussian'),
-
.
( , fspecial
,
, -
, ­ nrmx, nrmy).
imfilter (
conv ­ "convolution") -
. -
: 'symmetric'
-
, ' circular ' -
, ' replicate ' -
, ­ (
).
= /4, = 52 -
0
= 2 :
v a r p h i=pi / 4 ; omega=5sqrt (2) pi ;
omega0=2pi ; amv=omega0/omega ;
r d o t p h i=XXcos ( varphi )+YYsin ( varphi ) ;
f v=exp(- i omega( rdotphi ) ) ; fvz=fv . z ;
nrmx=length ( xx ) cos ( varphi )/( xx (end)-xx ( 1 ) ) ;
nrmy=length ( yy ) sin ( varphi )/( yy (end)-yy ( 1 ) ) ;
sigma=amv (nrmx+nrmy ) ;
g f= f s p e c i a l ( ' g a u s s i a n ' , round ( sigma ) , sigma ) ;
vw= i m f i l t e r ( f v z , gf , ' symmetric ' , ' conv ' ) ;
80

. 3.14. ,
-
. (
) .
­ , .. abs(vw), -
. 3.14.
, -
, -
.
.
-
, -
( = 10), -
a =
0
/ = 0.2 -
( -
). -
,
0
= 4 -
.
81

3.7
. 3.11,
. 3.14, ,
0.5, -
-
( . 3.11 ­ -
). ,
§ 1.4 -
, (-
) ,
- .
, , -
-
-
( -,
, -
), -
-
. , -
.
, § 2.2 ,
-
( ) -
,
. -
, , -
.
82

MATLAB
hilbert , -
. ,
,
-
,
( ).
,
,
- (-
,
).
:
function [ t i , f a ]= s A n a l y t i c ( t , f ) ;
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
f p ( 1 :N/2)= f l i p u d ( f i ( 2 :N/ 2 + 1 ) ) ;
f p (N/ 2 + 1 : 1 . 5 N)= f i ;
f p ( 1 . 5 N+1:2N)=flipud ( f i (N/2:N-1));
N=2N;
f a= h i l b e r t ( f p ) ;
f a=f a ( : ,N/4+1:3N/4);
fa
ta, -
, -
-.
-
, , -
. 3.15.
83

0
0.5
1
-2
-1
0
1
2
t
f(t)
-500
0
500
0
0.05
0.1
0.15
0.2
|F(omega)|
0
0.5
1
-2
0
2
-2
0
2
t
f(t)
H[f(t)]
-500
0
500
0
0.1
0.2
0.3
0.4
|F
analyt
(omega)|
. 3.15. : -
() -
, , ().
: ( ) .
, -
,
, -
-
. , -
, , -
. -
, -
- ( -
´ , -
, -
) .
, -
84

. 3.16. , (
) - -
0
= 2: ( )
( ).
, [ ta , fa]=sAnalytic(t , f );
[b,w]=fftMorlet(ta , fa ,a,omega0); ,
. 3.16.
. 3.16 . 3.11
,
,
/ -
-1 1, -
, ( ,
0
= 2 ) a = 0.05, = 40,
-
, .
--
, ,
85

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
0
1
t
f(t)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
0
1
b
Re[w(0.1,b)]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
0
1
b
Re[w(0.05,b)]
. 3.17.
-, ,
-. -
.
, -
, -
. -
, (
, )
(. (1.24) -
).
, § 1.5, -
, .
, - -
86

,
, : -
-, -
, ,
, -
-.
. 3.17.
-
,
(, ) . -
-
­ -
.
, , -
,
, . -
, t = 0 -
( -
a).
, -
, , t > a
. , -
, ,
.
, - -
, -
.
87
88 of 88 pages

Details

Title
Global and local spectral analysis of oscillating signals and images
Subtitle
Global'nyj i lokal'nyj spektral'nyj analiz ostsilirujushchich signalov i izobrazhenij
College
Kursk State University
Author
Year
2014
Pages
88
Catalog Number
V281539
ISBN (Book)
9783656841883
File size
3560 KB
Language
Russian
Tags
wavelets, Fourier transform, Morlet wavelet, spectral analysis, MATLAB
Quote paper
Dr. Eugene Postnikov (Author), 2014, Global and local spectral analysis of oscillating signals and images, Munich, GRIN Verlag, https://www.grin.com/document/281539

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