Спектральные представления сигналов

я
.
.
.
. .
-
2005
-2-
621.391.828
/
. . .
.- .
:
2005. - 34 .
"
,
"
"
,
"
"
".
,
,
.
. .
. .
. . .
, 2005
,
-3-
.
1.
Ч
я
1.1.
(1768 – 1830 . .),
21
1807 .
,
,
,
.
,
,
.
,
(t1, t2)
u1(t), u2(t),…, un(t), …
,
. .
t
0, i  j,

2

 u (t )u j (t )dt  k  0, (k  ),

t i
i
 i
1
i  j.
,
s(t):

s(t )  c u (t )  c u (t )  ...  cnu n (t )  ...   ci ui (t ),
11
2 2
i1
(1.1)
t
1 2
ci 
 s(t )ui (t )dt.
ki t
1
дciж
ciui(t)
,
s(t),
.
-4-
.
дui(t)},
(t1, t2), . .
:
t
2
 (t )dt   0, i  j,
u
t
u
(
)
k ,

j
i  j.

t i
 i
1
(1.1)
:

s(t )   ci ui (t ),
i1
(1.2)
t
1 2
*
ci 
 s(t )ui (t )dt, * ki t
1
.
,
,
.
?
,
.
,
(
.
)
,
(
)
,
.
,
,
.
(
).
-5-
дsin
nt,
(t0 , t0+T) ,
cos ntж
,
t0 -
T=2/ -
,
(t0 , t0+T)
s(t)

s(t )  a0   (an cos nt  bn sin nt ), t0  t  t0  T .
n1
a0, an
(1.3)
:
bn
t T
t T
t T
2 0
2 0
1 0
a0 
 s(t ) sin ntdt, n  1,2,3,... (1.4)
 s(t ) cos ntdt, bn 
 s(t )dt, an 
T t
T t
T t
0
0
0
(1.3)
Ф
(1.4)
,
:

s(t )   An cos(nt   n ), t0  t  t0  T ,
n0
(1.5)
An  an2  bn2 ,  n  arctg(bn / an ).
(1.6)
jnt
дe
(t0, t0+T)
Ф
(1.2)
s (t ) 
t0
} (n=0, 1,  2,…),
T=2/ ,
:
  jnt
, t0  t  t0  T ,
 C e
n n
(1.7)
t T
.
1 0
 jn t dt
Cn 
.
 s (t )e
T t
0
,
(1.8)
(1.3)
(1.7)
,
.
,
,
(1.4)
(1.8)
(1.3)
.
.
.
. .
1
Cn  (an  jbn ), a0  C0 , an  Cn  C  n , bn  j( Cn  C  n ).
2
(1.7):
(1.9)
-6-
s(t)
(t0, t0+T).
s(t)
.
,
(-, ).
,
,
:
s(t)
,
s(t) = s(t+mT), m=1, 2….
(1.7)
ejnt = ejn(t+mT).
T=2/),
(
Т
s(t) -
Т
s(t)
(1.3), (1.4)
(-, ),
,
(1.7), (1.8)
.
t0
1.2.
(1.8)
s(t)
Т
(
) Ancos(nt+n)
(n=1, 2, 3,…),
n,
=2/T),
( . .
n.
An
Ancos(nt+n)
.
–
,
 =,
.
2, 3, …,
. .
.
 = , 2, 3, …,
(
)
.
.
(1.7)
.
Cn e jnt (n  1,2,...)
.
Cn .
, 2, 3, …,
-7-
:
2 cost  e jt  e jt ,
–
,
.
,
.
Cn
(1.7),
,
n
An
(1.5).
.
j
Cn  Cn e n ,
:
,
,
Cn,
n
,
,
 =n (n=1,2,…).
(1.6)
(1.9)
Cn= An/2, n= n,
,
C-n= Cn, -n=-n,
,
.
–
,
.
s(t)=Acost + Bcos2t
,
,

2 .
я
1.3.
1.
,
. 1.

s(t)
A
-/2 0 /2
T
T
t
.1
:
-8-
 A,   / 2  t   / 2,
s(t )  
0,  / 2  t  T   / 2.
.
Cn
t0 = -/2:
 /2
.
A
1  / 2T
1  / 2  jn t
Cn 
s(t )e  jn t dt 
dt  
e  jn t | 
 Ae

T  / 2
T  / 2
jn T
 / 2
n  A sinn / 2 
2A
2A

(e jn  / 2  e  jn  / 2 ) 
sin


nT  2  T n / 2
2 jn T
A sinn / T  A sinn / q 


,
T n / T 
q n / q 
q=T/
,
,
=2/T .
(
s (t ) 
)
:
A  sin(n / T ) jnt
e
.

T n n / T
.
1
Cn  (an  jbn )
2
(1.10)
:
.
A sin(n / T )
A
bn  0, an  2 C n  2
, a0  C0 
.
T
n / T
T
:
s (t ) 
 sin(n / T )
A
[1  2 
cos nt ].
T
n1 n / T
.
Cn
(
.2).
(1.11)
-9-
Cn
C1
C2
C3
-4/
0
-2/
2

=2/T
4/
2/

.2
.
3
3
A sin(n / 2)
, n  1,2,...
An  2
T
n / 2
n()
.
sin(n / T )
n / T

e
 j 2k
 1, e
 j (2k 1)
 1 ,
k = 0,
1, 2, … .
,

T,
,
1.
An
)
A1 =2C1
A2
A
A0 
T
A3
0

2
2/
4/

n()
)
0

 2
-
-2
.3
- 10 -
«
2.
s(t )  A sin 1t ,    t   ,
.4.
S(t)
A
-2/1
-/1
2/1
/1
0
t
.4
T=/1 , =21 .
:
t0 =0,
A1  / 1
.
 j 2 nt
1T


jn
t
1 dt 

t
e
dt 
Cn   s ( t ) e
sin(
)

1

T 0
0
 j t
j t
1  j 2 nt
A1  / 1  j (2n1)t
A1  / 1 e 1  e
1
1

dt 
e
dt 
 e



j
j
2
2
0
0
A1  / 1  j (2n1)t
 j (2n1)t  / 1
A
1
1

e
dt 
| 
 e
2 j 0
2 (2n  1)
0
 j (2n1)t  / 1
A
A
A
2A
1



e
.
| 
 (2n  1)  (2n  1)
2 (2n  1)
0
 (4n 2  1)
: s (t )  
2A 
1 e j 21nt ,

 n 4n21
.5.
2A/
21 41 61
-61 -41 -21
0
-2A/3
-2A/15
-2A/3
.5

»
- 11 -
.
3.
,
Umcost,
U0 (U0<Um)
T=2  .
.6.
,
Umcos = U0, . . =arccos(U0/Um).
,
2
.
s(t)
2
Um
U0
-

0
2
.6
.6
:
s(t) = Umcos t – U0, - <  t <.
s(t)
1 
bn 
 s(t ) sin ntdt  0, n  1,2,...,
2 
:

s(t )  a0   an cos nt .
n1
:
Um
1 
(sin    cos ).
a0 
 (U m cos t  U 0 )dt 

2 
t
- 12 -
:
Um
1 
a1 
(  sin cos ).
 (U m cos t  U 0 ) cos tdt 
2 

n =2, 3, 4, …:
an
an 
2U m sin n cos  n cos n sin
.

n(n 2  1)
:
a0=Um0(),…,an=Umn(),
 0 ( ) 
0(), 1(),…, n(), …-
:
1
(sin    cos ) ,  1( )  (  sin cos ) , …,


1
 n ( ) 
2 sin n cos  n cos n sin
, n  2,3,... .

n(n 2  1)
(
)
,
,
,
.
,
Д2Ж.
1.4.
я
я
1.
: A=1 ,
 = 0,05 , ) = 0,1 , ) = 0,25 , ) = 0,5 , ) =1,0 .
2.
,
t
(
s (t ) 
.7)
:
A  sin(n / T )  jnt jnt
e
e
.

T n n / T
.
.
- 13 -
s(t)
A

t
t
T
.7
3.
,
«
»
. 8,
,
: s (t ) 
A



A
2 A  cos 2t cos 4t
cos 2nt


sin t 


...


...

2
  3
2
15
4n  1

S(t)
A
-2/
-/
/
0
2/
t
.8
,
(
2).
,
4.
. 9,
.
2
  sin(n / 2)  2
A A   sin(n / 4) 
n
2n
A
cos t ;
cos
t ; s2 (t )  
: s1(t )   A  
 


4 2 n1  n / 4 
T
2
T
n1  n / 2 
 1
A
2n
s (t )   A 
t.
sin
3
T
2
n1 n
- 14 -
S1(t)
A
-T/2
0
T/2
t
S2(t)
2T
A
0
-T/2
T/2
t
S3(t)
A
0
-T
t
T
.9
5.
s(t)
(
.
. 10).
S(t)
A



0
.10
.
s(t+) , . .

.

t
- 15 -
2.
.
Ч
я
2.1.
,
,
,
,
,
.
(-, ),
s(t),
.
S ( ) ,
:

.
 jt
S ( )   s(t )e
dt,

Ф
s(t ) 
(2.1)
.
1 .
jt d
,
 S ( )e
2 
(2.2)
ejt
. .
(-<  <) .
.
S ( )
1 .
S ( )d ,
2
(
)
.
(2.1)
,
(2.1)
(2.2)
,
(2.2) –
.
s(t)
S ( )
Ф
.
.
: s(t)  S ( ) .
,
.
S ( )
s(t)
.
.
- 16 -
(1.8)
(2.1)
,
.
,
,
.
j ( )
S ( )  A( )  jB ( )  S ( )e
,
(2.3)


A( )   s(t ) costdt, B( )   s(t ) sin tdt, S ( )  A2 ( )  B 2 ( ) ,


B( )
 ( )  arctg
.
A( )
S()
.
()
S ( )
( Ч )
(
s(t).
,
(2.4)
Ч
( Ч )
-
)
,
Ч
() -
(2.4)
S() -
.
(2.2)
:
1 
1 
j[t  (t )]
(
)

S
e
d


 S ( ) cos[t   ( )]d 

2 
2 
j 
1
(
)
sin[
(
)]



S


t


d


 S ( ) cos[t   ( )]d ,
2 
 0
s(t ) 
S()
(2.5)
().
.
Ф
.
Д1Ж .
1.
.
.
s (t )  S ( ) , s (t )  S ( ) ,
1
1
2
2
s2 (t )  s1(t  t )
0
.
.
 jt
0.
S ( )  S ( )e
1
2
S2 () = S1(), 2() = 1() - t0.
(2.6)
- 17 -
,
-
t0
,
(-t0),
-
.
j t
s (t )  s (t )e 0 ,
2
1
.
.
S ( )  S (   ) ,
2
1
0
,
(2.7)
0
. .
e
2.
j t
0 .
s (t )  s (kt) , . . k>1
2
1
, 0<k<1 -
.
.
1 .
S ( )  S ( / k ).
2
k 1
,
,
(2.8)
k
.
s (t ) 
2
3.
ds1(t )
,
dt
.
.
S ( )  j S ( )
2
1
(2.9)
.
1 .
S ( ) 
S ( ) .
2
j 1
(2.10)
t
s (t )   s ( x)dx ,
2
 1
4.
s(t)= s1(t)s2(t).
.
.
.
1 .
1 .
(
)
(
)

S ( ) 
S
x
S

x
dx
S
(
x
)
S

2
1(  x)dx .
 2
 1
2 
2 
(2.11)
,
1/2
.
- 18 -

s(t )   s ( x)s (t  x)dx , . .
2
 1
5.
s1(t)
s(t)
s2(t).
.
.
.
S ( )  S ( ) S ( ) .
1
2
(2.12)
s(t)
.
6.
–
.
,
s(t )   ri si (t ) ,
i
,
,
ri -
.
si(t)  Si ( ) ,
.
.
S ( )   ri S i ( ) .
i
(2.13)
7.
t
, . .,
.
s(t)  S ( ) ,
.
S (t )  2s( ) .
(2.14)
,
,
.
,

E   s 2 (t )dt .

.
(2.11)
s1(t) = s2(t) = s(t)
:
s(t)

1  2
1 2
E   s 2 (t )dt 
S
(

)
d



 S ( )d .
2 
 0

(2.15)
(
-
S2()
(2.15)
(2.15)
)
.
,
- 19 -
,
.
S2()
.
,

K ( )   s(t )s(t   )dt ,

S2()
:

 jt
2
S ( )   K (t )e
dt ,

K (t ) 
(2.16)
1  2
jt d
.
 S ( )e
2 
,
(2.17)
(2.1)
,
,

 s(t ) dt   .

,
,
-
(t)
,
.
,
,
.
2.2.
1.
я
(
.11 ):
 A,   / 2  t   / 2,
s(t )  
 0,  / 2  t   / 2.
(2.1):
 / 2  jt
.
A  jt  / 2
2 A j / 2  j / 2
S ( )  A  e
dt  
e
e
| 
(e
)
j
j

2


/
2

 / 2
sin / 2
sin / 2 .
2 A   

 S (0)
,
sin
  A
 / 2
 / 2

 2 
(2.18)
- 20 -

.
S (0)   s(t )dt  A .

. 11
S()
S(t)
)
)
A
A
-/2 0

t
/2
-4/
-2/
0
2/
. 11
1
.1.3
-
.
: s(t)=Ae-t1(t),
2.
>0,
1, t  0,
1(t)
: 1(t )  
0, t  0.
s(t)
A
A/e
t =1/
0
t
.12

 (  j )t
.
A
A
 jt
 jarctg( / )
S ( )  A  e t e
dt  A  e
dt 
e

. (2.19)
  j
2
2
0
0
 
(2.19)
Ч
S ( ) 
A
 2 2
Ч :
,  ( )  arctg( /  ),
- 21 -
. 13
)
.
)
S()
A/
()
/2
/4
A/2
-

0

- 0
-/4


-/2
.13
3.
(
)
(
s(t )  Ae
 t 2
.14 ):
,   0.







2

  t  j  
   t 2  j t
 (  t 2  j t )
.
2  
 2 4  
S ( )  A  e
e
dt  A  e
dt  Ae
dt 
e



A  2 4 
 x2 dx  A  e  2 4  ,

e
e




  x2
dx   .
: e

(
)
.14 ).
s(t)
S()

A

)
A
A 
e 
A/e
t
-1/
0 1/
-2
.14
0
2

- 22 -
4.
(
).
-
(t)
t=0,
,
1.
,

  (t )dt  1.

0
,
t
0
,



, t  0,
 (t )  
,
(t)
–
(
,
. 15 ).
, . .
(t)
,
(
,
)
.
:




s
(
t
)

(
t
t
)
dt
s
(
t
)

  (t  t0 )dt  s(t0 ),
0 
0

(2.20)
.

.
 j t
S ( )    (t )e
dt  e 0  1.

(2.20)
(t)
)
S()
)
1,0
1,0
t
0
0

. 15
(
.15 ).
(2.6)
 j t
0.
 (t  t0 )  e
,
(2.21)
:
 (t ) 
,
j 
1 .
1  j t
1 
1
j t
(
)
cos
sin




S
e
d
e
d


td


td







 costd .
2 
2 
2 
2 
 0
:
- 23 -
 (t ) 
1   j t
1

e
d


 costd ,
 0
2 
(2.22)
(t).
(2.22)
,
,
: s(t)=A.
  jt
.
1   jt
S ( )  A  e
dt  A2
dt  2A ( ).
e
2 

(2.23)
5.
,
. 16 .
(2.9),
s(t).
s(t)
)
A
-b
-a
0
b
a
t
ds(t)/dt
)
A/(b-a)
-b
-a
0
a
b
t
d2s(t)/dt2
)
-b
-a
0
a
b
t
.16
s(t),
:
d 2s
A
[ (t  b)   (t  a)   (t  a)   (t  b)].

dt 2 b  a
- 24 -
.
d 2s
 ( j ) 2 S ( ) ,
dt 2
.
( j ) 2 S ( ) 
.
S ( ) 
(2.21)
A  j b
j a
 j a
 j b 
e

e

e

e
 ,
b  a 
 e j a  e  j a e j b  e  j b 

  2 A cosa  cosb

.

 b  a
2
2
2
(b  a) 2 


2A
(2.24)
b =/2,
(2.24) a = 0,

:
2
.
4 A 1  cos( / 2) 8 A sin2 ( / 4) A  sin( / 4) 
S ( ) 


,

2   / 4 

2
2
. 17 .
)
S()
)
s(t)
A
-/2
A/2
0
-8/ -4/
/2 t
0
4/

8/
.17
.
6.
S ( )
:
1, t  0,
1(t )  
0, t  0.
1(t)
: 1(t )  1/ 2  (1/ 2)sign(t ) ,
 1, t  0,
sign(t )  
 1, t  0,
.
S1( )
,
sign(t),
(
.18).
–
- 25 -
sign(t)
1
0
t
-1
2(t)=dsign(t)/dt
2
0
t
.18
dsign(t )
 2 (t )
dt
,
(2.9)
(2.21),
.
.
( j ) S1( )  2, S1( )  2 / j.
(2.25)
(2.23)
(2.25),
1(t):
.
S ( )   ( )  1 / j.
,
(2.26)
1(t),
(2.26)
.
-
7.
Т.
s(t)
(1.7)
s(t ) 
s(t):
  jn t
,
 C e
n n
.
(1.8).
=2/T,
Cn

 .
.
 j t
 j t
S ( )   s(t )e
dt    C n e jn t e
dt 

 n
 . 1   j( n)t
.
e
dt
 2  C n

2

 C n  (  n).

2 
n
n
(2.27)
,
(n=0, 1, 2, …),
(
=n
,
)
.
2 C n .
- 26 -
(
(1.10),
. 1),
:
.
2A  sin(n / 2)
S ( ) 
 (  n) .

T n n / 2
(2.28)
(2.28)
.19.
S()
2A
T
-4/
-2/
0

2
4/
2/

.19
 j ( t  )  A j j t A  j  j t
A  j (0t  )
0
0.
 e e 0  e
e
s(t )  A cos( t   )   e
e
 2
0
2
2

,
.
A j
A  j
 (   0 ) 
S ( )  2 e  (   )  2 e
0
2
2
 j
j
 Ae  (   )  Ae
 (   0 ).
0
.
8.
S ( )
(2.29)
s(t),
:
x(t)
s(t )  x(t ) cos( t   ).
0
.
X ( ) .
x(t)
(2.11),
.
X ( )
(2.29),
(2.20),
:
.
1  .
 j
 j
 (   0 ) d 
S ( ) 
 X (   ) e  (   0 )  e


2 
.
.
1 j
1  j
 e
X (   )  e
X (   ).
0 2
0
2
(2.30)
- 27 -
,
(2.30)
,
x(t)
.
0
.
.
,
,
-
. 20 ,
)
.
X()
x(t)
-/2
0.
X ( )
0
/2
-2/
t
0
2/

S()=1/2[X(-0)+X(+0)]
)
s(t)=x(t)cos(0t+)
-0
/2
-/2

0
0
.20
.
X ( )
9.
x (t),
Т
x(t)
T(t):


x (t )  x(t ) T (t )  x(t )   (t  nT )   x(nT ) (t  nT ).
n
n
,
x (t)
(
,
 Т
(
.
.
x(t )  X ( ) .

 T (t )   (t  nT ) .
n
(2.27)
),
)
(
) x(t)
- 28 -

.
 T (t )  2  Cn  (  n),
n
.
1 T /2
 jn t dt  1 ,
Cn 
  (t )e
T T / 2
T
=2/T.
,
 T (t ) 
2 
  (  n)    ( ).
T n
(2.31)
(2.11)
x(t)T(t)
.
X ( )
():
.
  .
  .
X ( ) 
  X (   ) (  n)d 
 X (   )  ( )d 
2 
2 n  
1  .

 X (  n).
T n 
. 21
,
(2.32)
,
.
X()
x(t)
)
1,0
0
)
1,0
0
T
T(t)
t
x (t)
t
-
=2/T
0

 ()
0


X ()
)
1/T
T
-
t
0
0

 =2 
.21
(2.32)
.
X ( )
(
.
X ( ) ,
Т
.
=2/T)

- 29 -
2 ,
.
 -
X ( ) ,
.
X ( ) ,
,
.
X ( )
.
X ( ) ,
(
.21 ).
,
x(t)
x (t).
В.А.К
(
:
),
F = /2,
x(t)
x(t)
,
T(1/2F )=(/ )
,
2.3.
я
.
я
1.
,
.22.
s(t)

A
0
t
t
.22
-
2.
s(t),
.23.
-
.
- 30 -
S()
A
-
()=- t0

0

0

.23
3.
,
.24,
()=0.
S()
A
2
-0
2
0
0

.24
4.
,
.
,
s(t)  S ( ) ,
.
1
[s(t  T )  s(t  T )]  S ( ) cosT .
2
,
8,
.
,
t= 0.
s(t )  A(1  e t )1(t ).
5.
.
6.
,
:
 A cos t, t   / 2,
0
s1(t )  
t   / 2.
0,
) s (t ) 

 s (t  nT ),
n 1
) s (t ) 

 x (t  nT ) cos0t,
n 1
,

 A, t   / 2,
x1(t )  

 0, t   / 2,
0 >>2/T=,  <T.
- 31 -
7.
,
1:

1, t   / 2,
x1(t )  
0, t   / 2,

s(t )  A(1  m cos1t )  x1(t  nT ),
n
1</T = /2, <T , m1.
,
8.

,
,
,
(
)
,
.
9.
 A cos t,
0
 0,
t   / 2,
s (t )  
: ) 0 =20,
t  /2
) 0=2 (
).
10.
(
(n+1)
,
. 25.
s(t)
0
T
nT
t
.25
,
:
 j (n1)T
.
.
1 e
S ( )  S ( )
0
 jT ,
1 e
.
S0 ( ) -
11.
.
,
. 26:
),
- 32 -
s(t)
A
-

0
t
-A
.26
12.
: s(t )  A[1(t   / 2)  1(t   / 2)].
А
.
13.
 A cos t, t   / 2,
0
s (t )  
t   / 2.
0,
.
14.
«
,
»
. 27:
s (t)
A
0

T
T
.27
.
t
- 33 -
1.
. .
2.
.
. .
.
3.
.
.
.,
, 1986.
.,
, 1988.
. .
.
.,
,
1982.
4.
. .,
. .,
«
. .
».
5.
.,
. .
6.
.
.
. .
, 1972.
.,
, 1971.
.
.,
.
, 1987.
7.
.
. .
.
.,
.
, 1989.
.
1.
.
Ч
………………………………………………………………………..……...3
1.1.
………………………………………………….3
1.2.
…………………………………………………….6
1.3.
………………………...7
1.4.
………………………………………………12
2.
.
Ч
…………………………………………………..15
2.1.
………………………………………………...15
2.2.
…………19
2.3.
……………………………………………….29
…………………………………………………………………………...33
- 34 -
Уч б
ич к
-
би
. . .
603950,
,
.
.
60 84 1/16.
.
.
, 23.
.
. . 2,1.
№
.-
.
.
. . 2,3.
300
.
. . .
603600, .
,
.
№ 18-0099
, 37
14.05.01
.