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min




>0
,


2.5.1.




2.5.



(2.2.1 )).

. . f () 0 t

,

,

,

,

,

,

.

,

t .

,

,





,

,

f ()e t

f o () = f ()e t t

-t



88
-

< <

f () , " t

max

dt <

.

- t

0

.

"

(2.2.1):
(

f () e t

t .

e

- t

.

-t

(2.5.1)

,

,

:

e

- t

-

-

-

-

-

-

,
,


Fo ( ) =

-



f o ( )e


- j

d ,

1 f o (t ) = 2

-



Fo ( )e

jt

dt .

f () = t (2.5.3)

1 2 j

+ j - j



F (s )e s t ds
(



.

F (s ) = L[ f (t )s ] = L[ f ()] t

F (s ) f (t ) .

















- j

+ j

.






e
- t

1 f () = t 2



-



e

jt



d

-



F ( )e -(

+ j )

d

.

f () t
-




d = ds , j

1 f () = t 2



e (

+ j )t



d

-



f ( )e -(

+ j )

d

.

(2.5.2)

s = + j
-

F (s ) =



f (t ) e

- st

dt .

(2.5.3)

s

(2.5.2)

(2.5.4)
),

(2.5.4) ­

-

-

,

89




. 2.11.



. 2.10.



s, F (s

F (s

)

f () t









90

. 2.9,

)

.

j

f () t

f ()e t

.

. 2.9.

-t

,

,

(2.5.3)

( . 1.1).

,

Convergence).

ROC

, ..

,



­

max

,

> >

j ,

ROC

,

min

e

-

- t

(Region

f () , t

(

"

t < t0 ,

,

.

2.9).

s.

s,

of

"

-

-

.



, . ,


, ..

ROC

= Re (s ) ( Re ­

,I

­







f (t ) e

-t

,


), . ,

t > t1 , ROC


-

(

. 2.10, 2.11).



ROC







s.

, R OC





ROC


, .. -



s(

. 2.9).

F (s

)





. 2.12.

(

s,

m

)













(*)

f () - F (s t

)

(

).









f () t





, ..

0 ± j 0 (





(2.5.3)


. 2.12).



f (- t ) ­

= L[ f (- t ), - s ] + L[ f (), s ], t

F (s ) =





0

f (- t )e dt +

st





0

f (t )e

f () , t

- st

dt =

f () ­ t







.

,

t < 0,
:

(2.5.5)

91





-










.

F (s ) =





0

f (t )e

- st

dt

.



= ,







f 0 () = f ()e t t

- t


max







,



min

f () t

.

,

.

(2.5.6)

,

,











f () = e t

-t

(

>0

)

(2.5.6)



= - .







-

,







,

F (s ) .

(2.5.6)





f () t













f () Me t



mint

,

M



­

,

min

,

min

>

,

, ..









,

min

R e (s ) =

(

. 2.10),


min









o

e

-t

f (t ) dt M e -(

(2.5.5)





o

,

-

min

)t

dt = -

,

Me

-

-( -

min

min

min

L[ f (); s t

)t

0

=

M -

]

L[ f (- t );- s

min

.

:

F (s

]







.

s-





F (s ) ,

-

,

f (- t )e

t







=

.





,

max

,

s-







92

,

,

)

-

-

-

:



1
,
min


2.5.2.


( . 2.9).



(Res)

s

(2.5.4)

i

,

Res[

.

(s0 )]

1 f () = t 2j

Re (s ) = (

,

= lim (s - s

,

:

F (s ) .

(2.5.4)

,

- j

1 + j

1 - j



F (s )e ds =

) (s )



s s0

(s ) = F (s )e

0

n-

. 2.12).

st

,

F (s )e

+ j

,

=e

st

F (s

Res[F

s0t

i =1

m

.

st

s s0

lim (s - s0 )F (s

)

(2.5.4)

R.

(si )e

si t

]

,

s



)

0

,

Re (s ) =

, ..

f () t

m

(2.5.8)

(2.5.7)

s

93





0

-





F (s ) =
1 ­ s+a


f () = 0 t t < 0.
.








. Res[ (s 2.4.
0

F (s ) =

1 ­ s


.

)]

d n -1 1 lim (s - s = (n - 1)! ss0 ds n-1

.

s0 = 0 ,




:

Res[ (s

,

0

)]

=e

f () t


.



(2.5.7)

2.5.

Res[ (- a )] = e

f () = u () = 1 t t

Re (s ) = 1 > - a .

s0t

-at

1 lim(s - 0 )F (s ) = lim s = 1 s 0 s0 s

lim (s + a

Re (s ) = 1 > 0 .

[

t>0

)

0

1 =e s+a





:

f () = e t

- at





94

2.6.
s - a

t > 0.
s = -a ,

)n (s )].

-at

.

(2.5.8) :

(2.5.9)

(2.5.8)

(2.5.7)

:

-



2.5.3.




f () = -e - t u (- t ) - e t


Re (s ) = -2 . f () = e t


F (s ) =

1 1 1 ­ = (s + 1)(s + 2) s + 1 s + 2



s1 = -1





(2.5.7)

(2.5.7)

,

f () = - e t

,

,

s1

-t

(

-t

-e

s2 , . .

-e

-2 t

.

-2 t

, t > 0.

Re (s ) = -1 .

),

F (s

, ..

t 0.



)

max

< -2

min

> -1

F (s

2.6 ­


) () t


(2.5.7)
Re (s ) = -2 , s-

.

­

- 2 < Re (s ) < -1 .

.

,

Re (s ) = -1 .

.

-2 t

u () . t

:

F (s

f

)

s 2 = -2 .

f () t

.

min

95

-

-



t =0 ,
f (n ) () s n F (s t


,


.





H (s

)

G (s ) .

f () = h (t ) g ()dt = H (s )G (s t t

,

f () sF (s t

)





..

:

L[ f (t )] = e

- st

L[ f ()] = e t

f (t

n-


t



2.5.4.




)
o

96


Re (s ) = > ,

lim e st f () = 0. t

+s e





o





o

- st

- st

f ()dt . t

f (t )dt = sF (s ) - f (0 ) ,

)

)
f (0 ) = 0 .
(2.5.10)

F (s ) .

n -1
-



. ,
y () t


ak ­ f () t


y () t

Y (s

)

F (s ) ­


Y (s )


. (2.5.12) ,
F (s

H (s ) =



Y (s ) = F (s )

,

1


Y (s ) = H (s )F (s ),

k =0

N

ak s

k



k =0

,

N

,

a

k



k =0

N

d k y () t

)

dt

a k s k = F (s ) ,

k

(2.5.13)

(2.5.11)

.

= f () , t

.

(2.5.11)

(2.5.13)

(2.5.7),

,

.

(2.5.12)

(2.5.11)

f () t

.

f () . t

.

:

97





-

,



s = + i (
(2.5.13) ), (2.5.14) . , . . 2.12). , , : ,


(2.5.11) ,
H (s ) .

Ai ­

,

.

si ­


y () = t


-

98

):

s.

D (s ) =

,

,

,

i =1



k =0

N

,

si

N

H (s ) .

Ai e si t ,

.

ak s k = 0 .

y () t

,

(2.5.11)

.

,

H( s ),

f () , t

(2.5.15)

(2.5.14)

,(

(

:

.

-

-



,


F (s ) D (s ) ,
y () = 0 t


= t <0. 1 2j y () t
- j


,

y () . t

1 y () = t 2j

(2.5.16)

+ j

Re( s ) = >



+ j

- j

F (s ) st e ds. D (s )



1 Y (s )e ds = 2j

F (s ) . D (s )

y ()e t

st

min

,

,

-t

.

,

,

+ j

- j



H (s ) X (s )e st ds =

y () t

,

(2.5.7)

H (s

(2.5.16)

)




h() : t



),

h () 0 t

1 h () = t 2j

h ()e t

,

-t

t<0 (

+ j

- j



H (s )e st ds

h () t

,

.

(2.5.17)

.

,

99

-

,



. , ,
Xe


2.6. Z -


h () . t


1 D (s ) ,
R e (s ) =

Z-

.

(2.5.17)

j ,

min

­

.

.

2.6.1.



,

(

. 2.4.2 )





r

-n

(

z > 0) ,


,

r

­


n =-





100

z = re

- j arg z

= + j ,

j = -1.
j

( )=


,

n = -



f [n ]r

, f [n

f [n ]e

,

,

,

-n

,

- jn

]

<

,

f [n

,

]

,

(2.6.2)

(2.6.1)

.

-



(z =
-


.2.13. z = r1 -






z=
.


,



(2.6.1)







) z < R+ ( z = 0

)

)

z-

z = r2 .

r1 .

z-

,

0 R- < z < R + ;

X (z ) =

(2.6.3)

(2.6.3)

0 R- < z < R+

n = -





,

z >R

f [n ]z

);

-n

z-

,

(2.6.2).

u[n ] =



)

,

­

zz=0

( z = )

);

-

,

­

z

,

(

.

,

1, n 0,

z-

0, n < 0.

,

:

.

,

,

. 2.13).

z=0

(2.6.4)

(2.6.3)

z = r1 .

z

101

-



n = - n = -




z =e n = -







u[n ] r

-n

=





r

-n

< ,



..

,

n =0

z-





z=e

j

,

n =0

,

r > 1,

.

(2.6.1).





z = 1,





DTFT

arg z = = const :

F (z

)

j

=Fe

( )=

j

,



,

f [n ]e

,

- j n

.

­





z-



(2.3.11)

t = 1 .



F (s ) =

n = -





(2.6.3),







F (s ) ­ 2 -

F (s ) = F (z

)

f [n ]e

z =e

s

=

-n s

n = -



.



f (n )e

:

,

- ns

.

Re (s ) = ,







F (s + j 2 ) =

s = + j





f [n]e

-n (s + j 2

)=





f [n]e

- ns -2n

z = + j ,

















s



102

z = es .

e

= F (s

)

.

(2.6.5)

. -

.




,



si = i + j i s -







:

z

i = e i sin i .

i = e i cos i ,

zi = i + j i = e (

i

+ j

i

)

,

z = re

j arg z

,









zi :

ri = zi = i2 + i2 = e i ,

zi = i + j

(2.6.6 )



(2.6.6)





s-

0

arg zi = i + 2k .

2

z-









r0 = 1 ,

i = 0,

arg z

,

0













(

. 2.14).

i

j

- j







z-



.

s-

. 2.14.

z-

(2.6.6a)

+ j s -

i

2

103







z-

-
















s-

=0,







j .

(2.6.6)







0 > 0

0

2







r0 = e



0

> 1.

0 < 0







r0 = e

-

0

< 1.

,

s-





0 - j

0 + j ,









s-

,

z-





,

,

z-





,

,

-

z-





z-





.

,

s-

,

­







j (

.

.2.13.

),



,

.

ri < r0 .

(

.

,

(

(

)

(

(

)

,

.

.2.13. );

















­



2.7.

,

.

f [n ] = 1

.

n=0

f [n] = 0











.

-







104

)

,

)

.

,

,

s-

z-

-

, ,








­



[n ] =



z-

1, n = 0; 0, n 0.

.

(2.6.3),









F (z

)

F (z ) =

z-





2.8.

f [n ] = u[n ], . .



n = -





[n]z

u[n ] =



:

F (z ) =



n =0



-n

z

-n

.

= 1.

1, n 0; 0, n < 0.

:

=

1- z

1

-1

.

z-





z > 1, . .





,

,

(2.6.7)

:

,

(2.6.8)

(2.6.8)





z-



2.9.

.

a n , n 0; f [n] = 0, n < 0.



f [n

]

(2.6.3),



F (z ) =


n


=0

az

n -n

=

n (a z ) n
-1

=0

=

1- a z 1

-1

(2.6.9)
:
z = 1.

105

,






z > a,

F (z

)







z = a.

z-



):

2.10.

f [n] =



,

F (z ) =

n

= -

-1

(2.6.10)

u[- n - 1]z

- u[n - 1], n 0; 0, n > 0.

-n

=-

n



=1

zn =

(

1- z

1

-1

.

z < 1.







, F (z

)



,

z-

z < 1.

(2.6.8)

(2.6.10)









z-



2.11.

e j n , n 0; f [n ] = . 0, n < 0.



z-



F (z ) =



n =0



e

j n -n

z > 1,


n =0















106



z=e

j

.

z

=

.

(z


,

-1 j n

e

)

=

1 - z -1e 1

j

,

(2.6.11)

(2.6.10)

­

-

,



C n = - C




C C n = -





2.6.2.

z-

f [n

]







F (z ) .





.

­



,

z-

,

.

,

1 z 2j

.

k -1









(2.6.3)





z-





F (z

)



1 F ( z )z 2j

1 F (z )z 2j



k -1

1 dz = z 2j





,



f [n

k -1

]

1 z 2j







k = n, . .



1 z 2j



k -1

-n +k -1



f [k ] =

:

C

1 F (z )z 2j

dz =

C



dz =

k -1

1, k = n; 0, k n

dz



.



.

:



f [n ]z

- n + k -1

.

-n

.

dz

.

,

(2.6.12)

107





z-

-

-

-












,

(2.6.12)





F ( z )z

k -1



,

(2.5.8)

(2.5.9),







s





,

.

z.

,

.

F (z

-

)





,

z -1 .

,

,

(





p

z

F (z ) ,







M

i =1

i = 1 - pi z



(

i

(2.6.9)

-1

)F

(z )

i

z = pi

F (z ) =

F (z ) =

.



(1

(1

i =1

N

i =1 N

M

1 - pi z

- pi z

- zi z



i

-1

-1

-1

,

)

)

.

:

z-







i ( p

)n

.









108
i

:

:

(2.6.13)

.

) -



2.6.3.








f [n ] =

0, n < 0.



i =1

N

i (p

z-

i

)n

, n 0; .

z-





f [k ],

n

, n = 1, 2 ,

K





.

k

z-

k = 0, 1, 2, ...

ka

a

k

k

1

k

2

k

.

F (z ) =

(1

z

(1

(1

(1

-1

1 - az

-z

1- z

- az

-z

-z

az

(1

k =0

z

z

N

1

1

-n

-1

-1 n +1

+z

-1

-1 2

-1 3

-1 2

-1

)

-1

f [k ]z

)

)

-1

)

)

-k

(2.6.14)

109

.





z.


y [n

x[n

]

]

:

,

,

x[n ] y[n ] X [Z ]Y [Z ].

x[n - i ] z -i X (z

.

z -1 , z

: [n - i ] z - i .

-2


..



2.6.4.



a

bn ­







110
n

z-


N

k =o

a k y [n - k ] =
: ,

.

.

,

,


M

k =o

)

bk x[n - k
. ,

]

,

,

x[n

(2.6.16)

]

X (z ) .

,

(2.6.15)

y [n

.

]

-

-



H (z ) =
1 Y (z ) = X (z ) 1 - Az
-1


y [n ] = x[n ] + A y [n - 1] z-


Z-


z(2.6.16) (2.6.15)

2.12.

H (z ) =

H (z ) =

n = -

Y (z ) = X (z )

(2.6.13).





h[n ],

h[n ]z

1+

-n



k =o N

M



k =1

.

bk z

ak z

-k

-k

,

(2.6.16)





:

,

Y (z ) =

h[n ] = An ,

( y (- 1) = 0 ).

1 - Az

X (z

n 0.

)

-1

,

,

.

A < 1, . .





n


=0

A n < .

,

(2.6.17)

111



-



. (2.4.4) (2.4.5)


2.7.1.




2.7.



112

(decimating ­

(up-sampling)

x (n

).

)

,

N.

,

.

N­

.

.

y [n] = x[nN ] .

(

.

:

,

(

(down-sampling)

)

)

y [n

]

,

,




)


N Ye


Xe

()

jt





j

=X e

t

=X

1 = t t

k = -



2 -k . t t









()

j

=X

:

1 = Nt Nt



=-



2 -k . Nt Nt









Ye

Nt

k

X

X

k = nN + l ,





()

j

1 = Nt





N -1

2 n 2 l - - . Nt t Nt





t = 1 e

2n

Ye

=1

n = - l =0

n,


X



()

j

.

)

1 = N



N -1

k =0

X (( - 2k ) N ) .

. 2.15.

,

:

Ye

()
j

. 2.15

x[n ] ;

(2.7.1)

:

N = 3.

113





, 1 Y (z ) = N



N


.
. 2.16.

LP ­

. 2.5.1,

x[n ] :

,

;

,

,

,







N ­

N

h[n ] =

(2.7.1)

:

1 2

-





N

N

e

j n

d =

-

1 N

sin

N. n N

n

z-







e

+ j

z.

,

WN = e

N

"

- j 2 / N

.

. 2.16

,





X (W k =0

N -1

,

k N

z

1N

).

(2.7.1)


(2.7.2)



X ( ) =

x n = -


n,

.

' [n ]e

- j n

N

=

x n = -


(nNt )e

x' [n ] ,

- j n

1 = Nt

n = -


X -k

.

2 . Nt







114

.

"

x[n
:

1 , N

]

-

-





2.7.2.




­ , 1 X ' (z ) = N
x[nN


,


t = 1
:
,

X ( ) =

1 N



N -1

X -k

2 . N





,

z-

z-

, ..

(2.7.4) z

]

X (W z ).

N -1

k -0

k =0

:

y [n

k N

]

x' [n

]

x' [n ] .

y [n ],

(2.7.4)

(2.7.3)

z

1N

(2.7.1),

,






(2.7.2).

x[n ],

:

y [n ] =

x[n / M ], 0,

M

n = kM ,

k Z; .

M­

t .








z-

Ye

Y (z ) = X z
j

( ) = X (e )

()
M

jM

.

(2.7.6)

(2.7.5)

M

115



-

-

-

,



.. . n = kM , u[kM ] = y[kM ] = x[k ], , ,


. 2.17.
: LP ­ . 2.17.

,

,



M

h[kM ] = [k ] .


M. n M

. 2.18.

;M­

u[n ] = y [n ] h[n ] ,

h[n ] =

-

-

-

sin

n

(

M,

,

,

-

M

M

)







: LP ­



,


N

116



M



N

M-

2

.

,



296 . 236 . 4. 2. 3. 6. 5. . . .: .: . ., . , 1978. 847 . ., .. , 1983. . 1-2. , .. . .. . . .: .: . , .: . . . . , , 1989. 487 . ....., 1958.




, . 2.18.

1.

7. Vatterli M., Kova eve J. Wavelet and Subband Coding. Prentice Holl

c

c

PTR, 1995. P.488.

..

.

.:

, 1980. 381 .

, 1962.

117

-