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621.372.037(075.8)
32.811.173
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ISBN 5-98298-326-6
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621.372.037(075.8)
32.811.173
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.. ISBN 5-98298-326-6
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n f
(2.2)
A # * &, % &+ *# & " ; + # , # ' * & (2.1):
X # ( j Z)
f
³ x # (t) e
jZ t
f
¦ x (n T# ) e
dt
jZn T#
n f
f
f f
¦ ³ x ( t ) G(t n T# ) e jZt dt
(2.3)
n f f
& # +" + >@ * G- ' .
& % (2.2) (2.3) > + <& ( "# * &) % !#, &* &
>
# * E &
jZn T#
j(Z k Z # )n T# . $ e
e
# Z#:
$ $ F(jZ)=X[j(Z+kZ#)], k = 0, ±1, ±2,…( . 2.5). & $
"+ * $ . A #
%# # # & ( *&) . D E * > #+ % > $ * * ". D #> # * $ (0 ± Z#/2).
2.3.1. , A "+ %# # # # (2.1), # " >@ ' fG(t) # " >@ # ; + f G ( t )
x # (t ) x(t)
E
f
¦ Ck e
j k Z # t
.
k f
f
¦ Ck e
k f
' & #
24
j k Z # t
(2.4)
Ck
1
T#
n T# T# / 2
³
G( t n T# ) e
1 e jk Z # n T#
T#
jk Z # t
n T# T# / 2
1
T#
# % #+, > # ( * ) E * # * ' . ! * % &+ % & & " ; + @+> * # " >@* '
FG ( j Z)
1
T#
f
¦ G(Z k Z # ) .
k f
= " ; + (2.4) 1
T#
X # ( j Z)
f
Ak =1/!# f
¦ ³ x(t) e
jk Z # t
e jZ t dt
k f f
1
T#
# & % >
f
¦ X [ j (Z k Z# )]
k f
(2.5)
J % "+ # * # " >@* ' , >@ " # >
(2.1)
* :
X ( j Z)
f
1
2S
³ X a ( j - ) FG ( j (Z - )) df
1
T
f
¦
f
³ X a ( j - ) G (Z k Z - ) d-
k f f
1
T
f
¦X
[ j (Z k Z )]
k f
" (2.5) #, # +> #
% F(jZ), @&$ kZ#. = F(jZ) & kZ# &" % % jk Z t
# , >@ $ # " &$ E e
>@* '
fG(t) (. (2.4)
. 2.4). D
" G
# "' # "' .
& % (2.5) % &+ >
# " &$
# "'
&$ & ( &)
& , &
* > > > # + .
= &* * &
, & + * * Zm, >@* -
25
>: Zm<Z#/2. = E ( . 2.5) # * ±Z#/2 ( |Z|dZ#/2) # (# % !#) : !#F#(jZ)=F(jZ).
"' %# "#+ % . Zm<Z#/2
+ . E $ ' '"
@+> #+ ;?H + * * $ *
=(j Z), * !# |Z|dZ#/2 * > |Z|>Z#/2 ( . 2.5).
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|=(jZ)|
|X(jZ)|
Z
–Z#
–Zm
Zm
0
Z#
D Z#/2
–Z#/2
. 2.5. % ! ! !
" ˶ ุ 2˶ m
A &$ # ;?H " >
; + # # "
# =(jZ)F#(jZ)
x(t )
T#
2S
Z# / 2
³
3 ( j Z) X # ( j Z) e jZ t dZ
Z# / 2
f
¦
n f
(2.6)
x (n T# )
sin[Z # ( t n T# ) / 2]
Z # ( t n T# ) / 2
& % (2.6) " % x(t)
# " & >@ ' sinx/x & E ' x(nT#) ( %), &
@ .
% &+ # % * # $#(t) + * $ 26
* #+ ;?H h(t), " * & " ; +
* $ *:
h(t)
T#
2S
Z# / 2
³
sin[Z # t / 2]
Z# t / 2
3 ( j Z) e jZ t dZ
Z# / 2
(2.7)
f
= # (2.7)
x(t)
³ x # (W) h ( t W) dW
f
& # ' # (2.6).
H , # q#=2qm, " " % *
" ' E*.
+, # & # " * q#<2qm ( . 2.6) # * |q|uq#/2 # "' F(jq) "#+ # : T#F#(jq) v F(jq). =
& > > # @& F[j(q–kq#)] ( . 2.6 k=±1). < '? $ $ . A "& < # "' % "& > < . = % # & & .
T#|X#(jZ)|
|=(jZ)|
|X(jZ)|
Z
–Z#
–Zm -Z1
-Zc1
Zc1
0
Z1
Zm
Z#
D –Z#/2
Z#/2
. 2.6. % ! ! !
" ˶ <2˶ m
;* +* x(t) * # + T & , "$>@ .
27
! * % # + " # >@ * # + xw(t) + * * ' =!(t) * # & T: x(t)=xw(t)=!(t) ( "& &*
*
+ ). * E Xaw(jq) xw(t) * $ *
Xa(jq)=Xaw(jq)=!(jq),
=!(jq)=sin(qT/2)/(qT/2) * '
# "& > * # + . J , , >, * # &. ? % # ( . 2.7) > " & # "' , @&$ , & >@ * , +<> & # "' .
T#|X#(jZ)|
|=(jZ)|
|X(jZ)|
Z
–Z#
–Zm
Zm
0
Z#
D Z#/2
–Z#/2
. 2.7. % ! ! !
%
&* @+> #+ ;?H # + x * ( t )
N 1
¦
n 0
x (n T# )
' sin[Z( t n T# ) / 2]
Z( t n T# ) / 2
x(t)
> # + +.
H # "' * # + f#=2fm & N=f#Tc=2fmTc, "& " * x(t), "& > # * * fm, * " +>. D * * >@
x*(t), #>@
# $(nT#) * ±f#/2.
28
A % # "'
+&$ " % #& ,
"+ &* # * ±f#/2 % + * %
. J , & & >@ , % < <& $ q > q#/2
# "' > ">
>
# , "# $ % $ Zc Z k Z# d Z# / 2 .
@ & >@ >
$ # , #& > @ >@ , %
$ #&* . ? , F1 . 2.6 " Fc1=FGF1. , # Fc1 *
$ # &+.
# Fc1, " fA # fA " . 2.8.
fA
f#/2
fc1
fA
0
f#/2
f#
f1
3f#/2
2f#
. 2.8. < !
D + + $ % % , + # " , # *
# "' . E "> + ;?H1 $ # & DA, " – $'*.
2.4. +*(-((&( %%:9/ -%*()&'4 %%)(
* # > "& , @ >@
# &$ . & > E & # "' , +"> # ' * & '
* " # +>
+> .
, , +> #
$, >@ "+ %# &$ #&
$ #& # + :
29
y(n)=;[x(n)].
(2.8)
D &, "&* & &
# , "& > &.
=
# # & & ' > *&
*&, &
& .
\ *& # & & & ( E ' ) # > ' " ' :
(2.9)
y(n)=;[a1x1(n)+a2x2(n)]=a1;[x1(n)]+a2;[x2(n)]
( & "#* * %# "#* )
# , . . " % "#* :
y(nzm)=;[x(nzm)],
(2.10)
# x(nzm), y(nzm) z # + , "# %& ( # &
) + x(n) y(n) m # # "' !#.
2.5. ()-' )()(%0 +%&/ &(6&'4 -%*()&'4 %%)(
*((&&6 %) 0*)' 9:*6 :8)*9
& 4 %&(
& *& &
* & > #
' +& :
M
ª d k y( t ) º
a
¦ k « dt k »
¬
¼
k 0
y (t )
N
ª d l x(t) º
b
¦ l « dt l »
¬
¼
l 0
f
f
f
f
³ h(W ) x(t W ) dW
³ h(t W ) x(W ) dW , ,
(2.11)
(2.12)
# h(t) – + $ , >@ ' * *
& #+- +: h(t)=;[~(t)].
# &$ $ #
' + (2.11) +:
M
¦ a k y( n k )
k 0
N
¦ b i x (n i)
(2.13)
i 0
, #
' + (2.11), M "
# (M N), ak, bi – & E
' &; x(n z i),
y(n z k) z $ # *
&$ # * & &, "# %& i k
# # "' .
30
C" % &+ , , # "' * #
' + @+>
" #:
dxx(n)zx(nz1) – " + . #.
# #:
= a0 = 1 " (2.13) N
y( n )
M
¦ b i x ( n i ) ¦ a k y( n k )
i 0
(2.14)
k 1
*
, ">@ " (2.14) (. . & >@ " &
), "& > # & ' % (;).
A &$ # '
+ @ $ # x(n) #&#@ $ (N M) $ # &$ # x(nzi), y(nzk), " <&$ ( "&$) &"+ ' E
' ak, bi z ( . 2.9). " &, , # + , @* * % .
=
" $ E
' akv0 ; "& +' (C;). C " + "* ,
. . " &$ # y(n) #&#@ $ y(nzk) ( . 2.9).
x(n)
bN …
0 1
n-N
b2 b1 b0
n
n-2 n-1 n
y(n)
a2 a1 a0
aM …
0 1
n-M
n
n-2 n-1 n
. 2.9. < + !
" C" > (2.14) E
' ak=0 +'* * % (?C;):
31
N
¦ b i x (n i)
y( n )
(2.15)
i 0
J
+ " * " , &$ # * #
N #&#@ $ " < * bi * @
$ # ( . 2.9, $ * ). (2.15) "& >
% %@ + ( >@ " >@ % # " &).
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f
¦ h (m) x ( n m)
y( n )
m f
f
¦ h ( n m) x ( m)
(2.16)
m f
$ #@ (2.16) # ' h(m) ( h(n)) "& $+%* ,* * '. D # # * & '* $+%
u0(m)=1, m=0 u0(m)=0 m>0: h(m)=;[u0(m)]. " " * & h(m)=0 m<0 ( % %+ "#*); E A & # >
#:
y( n )
f
¦ h ( m) x ( n m )
(2.17)
m 0
" %& # # +&$ $ * % ( . 2.10).
u0(m)
h(m)
;: "*
h(m)
1
m
0 1 2
)
m
m
0 1 2 …
0 1 2 …
)
)
N-1
. 2.10. > " % ( ) " % % ?@A- () @A- ( )
> +> $ > +'
%', E $ "& > % H-%. = + * $ % # + *
C;. * 32
C; "$>@ + $ & % ,
f
¦ h ( m) f .
m 0
E+' ' %' H%, . . +
* + * $ *. &% A (2.17) # ?; & #& ,
#& # * + * $ N:
y( n )
N 1
N 1
m 0
m 0
¦ h ( m) x ( n m)
¦ h ( n m) x ( m)
(2.18)
J ", BC % # +" + "' ?; , " % #
C; # $ + * $ " E +< G & *.
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$ & % A (2.18) %# & E
' bl
" ?C; (2.15): h(m)=bl|m=l >, " , E
' ?C;. = E , & +< " + # &$ , +, ?C; "> *
% BC (2.18), C; – " @*
(2.14).
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%%)( & +(%&6 +%%) ( %))&6 %))
? @ # &$ $'*
$ * :
f
X a (s)
³ x(t) e
st
dt ,
(2.19)
0
# s = +jq z &* \ .
= " > \ * jq ( ) " ; + , #>@ :
X a ( j Z) X a (s) s jZ
f
³ x(t) e
jZ t
dt .
(2.20)
0
# &$ " \ "
(2.19) @+> ": t nT#, , dt 1, . . # "' *
(2.19) :
33
X(s)
f
¦ x (n ) e
s n T#
.
(2.21)
n 0
D#
&$ " (2.19) # ' +& ' # &$ * S- .
J %*< > Z-$" ',
, # < f
¦ x (n ) z n .
Z{x (n )} X(z)
(2.22)
n 0
Z
(2.22) " * S ( -
s T
V T
jZT
#
\): z e # a j b e # e
.
J "+ > % " * S > Z- + ( . 2.11). D + E $
% * ">
$ ' * q, -
* # +> * E & e
z
e
VT#
e
j[Z k Z# ]T#
jZT#
k=0; r1; r2
S- +
# q#:
. #.
Z- +
jZ
jb
3Z#/2
Z#/2
-V1
Z1
0
e
V
Z#/2
Z1T#
-1
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+Z
Z1
V T
1 #
1
a
0
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. 2.11. S- Z- %
34
!, * jF S- (=0) ' % + # # Z- ; %# * <
* q# E # $ # E * % .
D# " % * ±q#/2. \ S- + ( < 0) & + # # Z- , S- + ( > 0) % " #&.
Z- " , & # * % , # $"? +% (2.2), #>@ :
X(z) z e jZT#
f
¦ x (n ) e
X( j Z)
jZ n T#
.
(2.23)
n 0
A# +, Z- " , " \, % &+ – # # + * x(n) = 0
n < 0, +, x(n) v 0 n < 0; E #& n zf # +f.
A * Z- " :
*%:
(2.24)
Z{a1 x1 (n ) a 2 x 2 (n )} a1 X1 (z) a 2 X 2 (z)
(Z- " & Z- " *);
:
Z{x (n m)}
f
¦ x ( n m) z ( n m ) z m
X(z) z m
(2.25)
n 0
(Z- " "# % m # x (n m) " # > Z- " X(z) "# % $(n) % + "# % z m . J G +" z m
# " E "# % &$ $$ DA:
, E "# % # x(n)
x(n-m)
–m
, . . # # "' , "z
-m
1
X(z)
z X(z)
z : Z{x (n 1)} X(z) z 1
c:
y( n )
f
¦ x1 ( m ) x 2 ( n m ) .
x1 (n ) * x 2 (n )
m 0
Y(z)
f
f
¦ ¦ x1 ( m ) x 2 ( n m ) z ( n m ) z m )
n 0m 0
35
X1 (z) X 2 (z) (2.26)
(Z- " # $ # + * > Z- " * E $ # + *);
$:
1 X (-) X (-) dy(n ) x1 (n ) x 2 (n ); Y(z)
2
³ 1
2 S j
-
C
" #-
(2.27)
(Z- " " # # $ # + * * Z- " E $ # + *, # - – , A – , $ & >@ * & #& + * ' ).
? * (2.27) Z- " " # # &$ # + * # "& % # &, >@ % &, # &$ :
f
¦ x (n T# )
2
n 0
1 X ( z) X (z 1 ) z 1 dz
2 S j ³
C
T#
S
Z# / 2
³
2
X( j Z) dZ . (2.28)
0
D " % * * K .
& * # " ; + # .
D & Z ; + " #> & % :
x(n)
T#
2S
x (n )
Z# / 2
³
X( j Z) e
jZ n T#
dZ
Z# / 2
1 X ( z) z n 1 dz
2 S j ³
C
1
2S
S
³ X( j O) e
jO n
dO (2.29)
.
(2.30)
S
¦ resi [X(z) z n 1 ] z
i
z pi
U#+ "&: =q#=2f/f# – + , "& % '
* *; res – && #& + * n–1
' F(z)=X(z)z
&$ $, $ & &$ A, # . # - ' +&$ ' *
X(z)=P(z)/Q(z) & > Q(z),
"& & > zpi '
X(z). = >& &+ @& , - %& , &
& . &&
>$ $ # @+> & % *:
# >
resi >F(z)@z z
lim z o z pi (z z pi ) F(z) ,
(2.31)
>
pi
# > +> r
resi >F(z)@z z
pi
@
>
@
d r 1 ( z z ) r F( z) .
1 lim
z o z pi r 1
pi
( r 1)!
dz
36
(2.32)
=
n=0
res 0 >X(z) / z @z 0
+ 1 / z z n 1
(2.30) # # +&*
&
lim z o 0 >X(z)@ > zp0=0, &* # % -
n 0
.
A@ > ' +& '& &$ Z- " *
# <
# &$ ' *.
P(z) # - ' + * ' , & >@*
X(z) (P(z)=0), "& > . D % # * > * Z- +" * ' Z- " # &$ .
2.7. (%)'( +%(-)(8&%) -%*()&'4 %%)(
=
# & # & # + +">
$ DA, $ &+&$ .
­1, n 0
1. '* $+%: u 0 (n ) ®
¯0, n z 0
&* &* U0(jq)=1. Z- " U0(z)=1
# &$ % " , #+- +
# &$. D $+% , # * &.
2. '* $+%, '* m :
­1, n m;
u 0 ( n m) ®
¯0, n z m.
A * "# % Z
F (; +)- "&
- jZ m T
#
Z{u 0 (n m)} z m ; F{u 0 (n m)} e
.
A @+> u0(n – m) > # # + + %
&+ # # # * :
f
x (n )
¦ x ( m) u 0 ( n m) .
m f
­1, n t 0;
3. '* : u1 (n ) ®
( ' > ).
¯0, n 0,
Z- " U1 (z)
f
¦ zn
n 0
1
1 z 1
-
* . A # > zp=1 + z0=1 # ( . 2.12, ).
37
& % # # $ # #>@ " :
1
U1 ( j Z) U1 (z) z e jZT#
1 e
jZT#
e
jZT# / 2
1
[e
jZT# / 2
e
jZT# / 2
]
j( ZT S ) / 2
#
e
2sin(ZT# / 2)
(
J* : e r jD
& # +" 1
2 sin(ZT# / 2)
U1 ( j Z)
# cos(D) r j sin(D)
" . 2.13,. D u1(n) $, , # * &.
4. &+%'* $+% * %:
­1, 0 d n d N 1;
u N (n ) ®
¯0, n 0, n t N.
Z- " U N (z)
N 1
¦
1 z N
1 z 1
zn
n 0
* * , #
uN(n)
Z- ":
. E % "+ %
# uN(n)=u1(n) – u1(n – N), N
U N (z) U1 (z) z N U1 (z) 1 z 1 .
1 z
A # >
zp=1
, i=0, 1,…N–1, (1 z ) 0, z 0i n 1 e
# * % ( . 2.12,).
A + # & % N
U N ( jZ)
1 e
j Z N T#
1 e
e
N
*:
"@&$ j2Si / N
e
j Z N T# / 2
j Z T#
e
[e
j Z T# / 2
j Z N T# / 2
[e
j Z T# / 2
e
e
j Z N T# / 2
j Z T# / 2
]
]
jZ( N 1)T# / 2 sin(ZNT# / 2)
sin(ZT# / 2)
# U N ( j Z)
sin(Z N T# / 2)
sin(ZT# / 2)
. 2.13,. D # * '
#>@ " $ &$ $:
­ N, Z 0;
U N ( j Z) ®
¯0, Z 2 S i / N T# i Z# / N,
38
sinc
" -
i 1,2,...N - 1
jb
jb
Z02
Z01
Z0
-1
Zp
1
0
2S/N
Z00
a
0
-1
a
Zp 1
Z0(N-1)
)
)
jb
jb
Zp
Zp1
Z
a
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Z
ZT#
a
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1
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Zp2
)
!)
. 2.12. " + ! ( ), " % % (),
( ) (!) 4 # " $ #&$ '
# $ # &$ $.
-
5. $'* * " qc:
x (n ) e
jZc n T#
Z- " , X(z)
cos(Zc n T# ) j sin(Zc n T# ), n t 0 .
@ , # * * :
f
¦e
jZc n T#
zn
n 0
A zp
e
jZc T#
# + z0=0
qc (
1
jZ T
1 e c #
z 1
.
# &* >
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jZT#
1
1 e
jZc T#
39
e
jZT#
e
j[( ZZc )T# S ] / 2
2sin(
( ZZc )T#
)
2
.
|UN(jZ)|
|U1(jZ)|
1/2
Z
Z
Z#
Z#/2
0
-Z#/2
-Z#/2
)
|X(jZ)|
1/2
Z
Z
1/2
Z#/2 Z#-Zc Z#
Zc
0
Z#/2
)
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0 Z#/N 2Z#/N
-2Z#/N -Z#/N
-Z#/2 -Zc
0
)
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Z#/2
Z#
!)
. 2.13. < " ! ( ), " % % (),
( ) (!) K #+ 1
X( j Z)
( ZZc )T#
2 sin(
)
2
# qc= q#/4 " . 2.13, .
6. @'* * " qc:
e
x (n ) cos(Zc n T# )
jZc nT#
e
2
jZc nT#
, nt0
Z- " X(z)
1/ 2
1 cos(Zc T# ) z 1
1/ 2
1 2cos(Zc T# ) z 1 z 2
r jZc T#
- %&$ > z p1,2 e
1 e
* z 01 0
jZc T#
z 02
z 1
1 e
z 1
cos(Zc T# ) (
A X( j Z)
jZc T#
. 2.12, ).
X(z) z e jZT# , "&* #>, "
. 2.13, # qc= q#/4.
40
' 5,6 +"> # " $ #&$ " + &$ # &$ .
'
7. $'* $+% * # + NT#:
x (n ) e
jZ0 n T#
Z- "
cos(Z0 n T# ) j sin(Z0 n T# ), 0 d n d N - 1.
f
¦e
X(z)
jZ0 n T#
jZ NT
1 e 0 # z N
jZ T
1 e 0 # z 1
zn
n 0
; +- " X ( j Z)
1 e
j (@ @0 ) N T #
1 e
j (@ @0 ) T #
e
j
0
2
(N 1 )T
sin (
0
N T )
2
0
sin (
2
T )
.
A #
+ # + 4, @ q0 ( " " & q0).
>
8. @'* $+% * # + NT#:
x (n ) cos(Z0 n T# ), 0 d n d N - 1 .
Z- " ; +- " &+ # & -
* Z- "
N 1
X(z)
¦
n 0
X( j Z)
e
#
jZ 0 nT#
jZ nT
[e 0 # e
] n
z
2
j
Z Z0
( N 1) T#
2
2
+
e
jZ0 n T#
e
jZ0 n T#
ª 1 e jZ 0 NT# z N 1 e jZ0 NT# z N º
«
jZ T 1
jZ T 1 »
¬ 2[1 e 0 # z ] 2[1 e 0 # z ] ¼
Z Z0
NT# )
2
Z Z0
sin(
T# )
2
sin(
Z Z0
( N 1) T#
2
e
j
2
Z Z0
NT# )
2
Z Z0
sin(
T# )
2
sin(
.
# % + @ @&$
# + + q0 – q0 .
2.8. (*(-)&/ :&9/
%))&/ 4*)(*%) -%*()&6 %%)('
= # ' * & # < " * \ &$ # $ # : H(s)=Y(s)/X(s).
D # & >@ & #
' + (2.11)
(2.12) # @ & % # # &$ ' * &$ # # - ' + * ' * * S
41
N
H(s)
B( s )
A (s )
¦ b i s
i
¦ a k s
k
i 0
M
(2.33)
k 0
# " \ + * $ & h(t)
f
H(s)
³ h(t ) e
s t
dt .
(2.34)
0
(s)=0 " (s)=0 # * ' (2.34) > s0i > spi &, " & # ' # "& *
+- > *
:
M
H(s) C
i 1
s s 0i
,
s s pi
# C – >@ .
= # * '
$ # $ H( j Z)
(2.35)
* & # Y ( jZ)
X ( jZ)
H(s) s jZ ,
(2.35) & % ; +- " + * $ H( j Z) H(s) s jZ
f
³ h(t) e
jZ t
dt .
0
= # * ' * # * & "& < Z- "
&$ # $ # &
Y(z)
.
X(z)
H(z)
& & % , & Z- " " &$ * (2.13), (2.14) (2.15), > #
# # - ' + * ' * ' + * ' *. "+ < " # * & "
& < $ *, & >@ $ # > ' > ( # < #
' +&$ * # &$ ).
" Z- " &$ &$ * A (2.16)
f
Y(z)
f
¦ ¦ h ( m) x ( n m) z n
n 0m 0
42
H(z)X(z)
#, # ' # * & Z- " + * $ f
H(z)
¦ h ( m) z m .
m 0
+ $ & ,
#+, Z- " > # * '
h (n )
>
-
1 H ( z) z n 1dz .
2Sj
C
³
H $ # * &, # < ; +- "
&$ # $ # , " %# Z ; +- " (2.12) $ # # * '
& H(z) * " * z e
H( j Z) H(z) z e jZT#
jZT#
:
Y ( jZ)
.
X ( jZ)
\ " +, ; +- "& # &$ , & $ # * & # > *
'
+ * * & =qT#=2f/f#, "& * % '
* *. U * & q
* #$ (0..q#) (zq#/2..q#/2) > " '
& #$ (0..2 ) (z..). H $ # *
& ' '
* & # & % :
H( j O) H(z) z e jO
H( j O)
Y ( jO )
,
X ( jO )
f
¦ h (n ) e jOn .
n 0
U# " $ " " & # "' q#, >@ * +
$ # # "' !#=1.
= "
@ #
" % # * $ # < &$ " * &$ # $ # &$ $ & < % :
H( j Z)
Y(n )
X(n )
X(n) e
jZ n T#
.
K #+ * $ # * &
#> * * * 43
"& > # - * (HF)
" * (;HF) $ &.
J % + * * * * $ , " &$ * ' *
&, – *.
" # * $ # * & " ; + + * $ # % *# q# 2 ( . 2.14). D
# @ ; +- " >&$ # &$ # + *, # &$ .
K E
G # +> *
E & e
jZn T#
H( j Z)
e
j(Z k Z # )n T#
f
¦ h (n ) e
e j(O 2Sk )n
j(Z k Z # )n T#
H[ j (Z k Z# )]
n 0
H( j O) H[ j (O 2 S k )] , # k= 0, ±1, ±2,… .
|H(jZ)|
1
0.5
Z
Z0
Z#/2
Z#-Z0
Z#
Z#+Z0
. 2.14. # CA ! ! %
= " " '
&$ +
& # # $ * $ * 0 # ±q#/2.
! '
&
+ $ $ $ #, # &$ + , + # +&$ % +&$ , # # " 0..q#/2.
U + * $ '
+ & # "' , %# * * # "'
+ * $ + , @ # * % * #
+> &$ * $ + . " & # "' q# q'# " < * $ 44
H(j q) = q'#/ q# ", ' + % > $ '
+ (E # " * " ; +). H * $ H(j q')=H(j q) > & " &$ ' +&$ '
+ q'i, "& $ $ #&
" qi ( , q0 . 2.14) < q'i= qi.
A# + , " & # "' $ '
+ " > & # "' .
! " , $%, % ' '* ,* $ * ', '* F/2, ' + > & $ '
&$ +
# &$ &$.
2.9. (*(-)&'( :&9 *(*%&'4 9:*'4 :8)*.
%( :A(%6 *(A(%)
C &* + (C;) " , &&* " & (2.14). & Z- " &$ &$ * (2.14), & * * "# % Z- " (2.24), (2.25):
Y(z)
N
¦ bi z
i
i 0
M
X(z) ¦ a k z k Y(z) .
k 1
= $ # < > Y(z)/X(z), @ & % #
# * '
+ :
N
H(z)
Y(z)
X(z)
1
B( z )
1
A(z )
¦ b i z
i 0
M
i
1 ¦ a k z
k
.
(2.36)
k 1
D # < # $ + % " * * z–1. % –1
# # $ # &$ ' *: H?(z)=B(z ) – * + , . .
+ HP(z)=1/A(z–1) – * H(z)=HH(z)HP(z).
& % # * '
(2.36) + –1
#
* z , , &< " , % "# % # # "' , # , %
+ +
+ >@ # * '
', #>@ .
45
? , # * ' H(z)=b0/(1+a1z–1+a2z–2) " y(n)=b0x(n)Ga1y(nG1)Ga2y(nG2).
D # , + %
+& , " (2.14)
# * '
(2.36), % # " + & % # # *
' & " " >, , &<, Z- " >.
& %
(2.36) # * ' , " (2.14), # , # N B(z–1) &< # M A(z–1), #>@ # '
+ K, . . & + N P M. N > M # ' (2.36), " >
(2.14), # "+ # # $ # &$ ' *, " &$ ( ) # * ' + (N z M)- #, ( # ) # * '
+ ,
# * # +< K.
" '
&$
+
# > ' > (2.36)
=
& %> %
# < B(z) A(z) % +& * Z. E # +
"+ (2.36) % + zM
N
H(z)
B( z )
A(z)
zM N
¦ b i z
i 0
M
N i
1 ¦ a k z
M k
.
(2.37)
k 1
K % + zM–N N < M " (M z N) &$ * ( N < M (N z M) &$ * "), &, , > *
" (. . * > ).
D# # ' + % &+
+ " Z- " " , # < # $ # "# "
% +& * Z:
N
H(z)
¦ b i z
N i
¦ a k z
M k
i 0
M
.
(2.38)
k 0
# & N u M "+&. H & # +
E , # +
"+
M
(2.38) z
46
H(z)
b 0 z N M b1 z N M 1 ... b N M z 0 b N M 1 z 1 ... b N z M
1 a 1 z 1 ... a M z M
. (2.39)
& % > (2.39) #>@ " :
y(n ) b 0 x (n ( N M )) b1 x (n ( N M 1)) ... b N M x (n ) b N M 1 x (n 1) ... b N x (n M )
>a1 y(n 1) a 2 y(n 2) ... a M y(n M )@ .
= E > @ * &$ # y(n) & + @ x(n) #&#@ x(nzi) $ # , (NzM) #@ , . . ' x(n+NzM), x(n+NzMz1), … x(n+1), " % . ! + # *
" , + NPM, ' , "# *
(2.38), &< + ",
# " + * + # * & ( &$ # * # % %+ $ # *).
& % (2.38) N < M, >@ " # + #:
y(n ) b 0 x (n (M N)) b1 x (n (M N 1)) ... b M N x (n ) b M N 1 x (n 1) ... b N x (n M )
>a1 y(n 1) a 2 y(n 2) ... a M y(n M )@ .
U#+ & $ # x(n), x(n z 1), … x(n z M+N+1) +"> & y(n), " ""#& > + (M z N) .
! " , # & '
+ (2.37), (2.38) &+ @ # & < # $ B(z) A(z) # # N=M:
N
H(z)
¦ b i z
N i
¦ a k z
N k
i 0
N
B( z )
A(z)
(2.40)
k 0
= # & ' (2.37), (2.38) N < M # (2.40)
> E
' bi N < i uM.
#&$ &< $ + *
& # * ' E
' & b0 a0 ( i=k=0) >
< >@> + " @ &+ %&
& # '.
= # * '
(2.40) $ # >,
>@ > + " " &$ + .
47
E+ # * ' > (2.40),
. . " * Z, @>@ + B(z):
B(z)=0 z=z0i.
&? > " # * ' :
A(z)= 0 z=zPi.
& + & @ & E
' ak, bi >
@ & (
&)
/ %& >.
?
> &+ &
& .
# # * ' N +< #
" K, *
+ (M z N) *, &$ >,
. . "@&$ # * Z- .
U >& # * ' , % # +
"& * +- > *
, +" " % " E & % :
N
H(z)
i 1
N
( z z 0i )
( z z pi )
(1 z 0i z 1 )
(1 z
i 1
1
pi z )
.
(2.41)
& % (2.41) % + >@ * % ' & b0 a0 # * ' + C=b0/a0, E
& # '.
& % # # * '
(2.41) +"
"'
&$
+ . , @ $ & % * +&$ $ &$
+
$ # & ' .
# ' "#
# & E &$
# * (2.42)
H(z)
r
D( z )
r
( z z pr ) ( z z p ( r 1) )...( z z pM )
Bu
¦ (z z
u 1
pr )
u
M
¦
i r 1
Bi
B0 ,
( z z pi )
$', $? & % # + * $ E #
(2.42)
M
h (n )
¦ Bi z npi i , n>0; h(0)=B0.
(2.43)
i 1
$? r
& % # + *
= $ E # > E
' &
h r (n )
Br
(n 1) (n 2) ... (n (r 1)) z npr r , hr(0)=0.
( r 1)!
48
(2.44)
! " , & % (2.43), (2.44) # E &$ # *
&
& > " > # # + +> $ + # * ' , "# *
(2.42).
2.10. %))&'( 4*)(*%) *(*%&'4 :8)*.
%( %)6%)
H > $ C; % + > "
& % * # *
'
H(z) – # + (2.40), (2.36), +- > (2.41) " % jZT
#.
E & # (2.42), +" " z e
= # * ' (2.36) $ C; #
N
¦ b i e
H( j Z)
i 0
M
-
jZT# i
1 ¦ a k e
jZT# k
.
(2.45)
k 1
= # * ' C; +- > *
(2.41) & % # HF, "& >@ & :
H( j Z)
M e j Z T# z e j Z 0i T#
0i
i 1e
j Z T#
z pi e
M R ( jZ)
R 0pii ( jZ) .
i 1
j Z pi T#
(2.46)
H + "+ (2.46) # > & R0i, Rpi # * % * # *
qT# # * z0i > zpi C;. = $ # #>
(2.46) # & % # HF ;HF C;
#
H( j Z)
M R ( jZ)
0i
R
i 1
M(Z)
pi ( j Z)
,
(2.47)
M
¦ \ 0i \ pi ,
(2.48)
i 1
R0i(j q), Rpi(j q), 0i, pi – &, # |R0i|, |Rpi| z # & "& +> ' .
C % > * Z- ( . 2.15)
"
#
& , # >@ $ $ * # * % , >@* "# * qT# (49
, * + HF
;HF
# " (0..q#/2).
. 2.15), @+> (2.47), (2.48) % &+ # >&$ " * & jb
Rp1
Zp1
Z02
-1
A(Z1)
R01
R02
Z01
1
0
a
Rp2
Zp2
. 2.15. " + ,D
! * * + # # * '
# * $ &$ * C; " * > . (2.47)
. 2.15 q=qpi
+ # > Rpi.min > +&
" HF E
' # C;, q=q0i + # * R0i.min – +& " E
' #
+ .
! " , $ +* $ $+ $$+ * $ + , $ $? – $+ .
. 2.1 #& " HF C; 2-
#, & & * > ( . 2.15) # #
&$ '
&$ . D %& #
> > " z01=1, z02=z1, zp1=0.4+j0.6, zp2=0.4–j0.6. =
HF * + % - >@ + .
! ' 2.1
* CA ,D 2-! ,
" +
=qT#
()
0
0
1.5
p1
4.82
/2
2.24
0
C # %& " > *
Z- " $ " , % "
+ C; "# * * $ *, " + #.
50
A @+> *, "@&$ # * % , E > & "$ HF.
?, &<>@ #> # ', > C; + - " , +"& #&$
" &$ .
= * > * # % *
C;.
* +*.
= > *
C;, &<>@ #> # '
(|zpi|<1), $ # # # . *
+ > $ # * S- , ( . 2.11) Z- " % + # # .
2.11. !*' *(A9 *(*%&'4 :8)*
C &
+ & @ > " & . D &
$ "' >
x ;
x ;
x #;
x +.
C" & >> & +& ' % , "# % .
? &$ $$ * "' ; > E& '
* "# % (Zz1), % (F) & ().
C; # &< ( >
# K @ " (2.14)) ">, , # + + # " &$ " + #, # * ( # + *) + *
$ "' .
" & # & " + C; & > * *
"' C;.
H # +&$ +&$ " + L #
+ M $ # L=K/2, – L=(M+1)/2, E # " + " # (" # @ & > ).
51
++.
= # ' # " C; ( . 2.16) # " # # &$ ' * " + :
L
H(z) C 0 H J (z) .
(2.49)
J 1
1 b1J z 1 b 2 J z 2
H J (z)
1 a 1J z
1
a 2J z
2
z 2 b1J z b 2 J
2
z a 1J z a 2 J
( z z 01J ) ( z z 02 J )
,
( z z p1J ) ( z z p 2 J )
# HJ(z) z # ' J- # " "
E
' b0J= 1; A0 – >@ * < >@ *
% +.
x(n)
H1(z)
H2(z)
HL(z)
…
y(n)
. 2.16. " " ,D
" # E
' & b2J a2J (2.49) &
>. $ #& J- " xJ(n) * &$ # * yJ–1(n) #&#@ (J z 1)- " : xJ(n)=yJ–1(n).
E
' & " +
$ >, "& < :
z 01,2J
b1J
b1J r b12J 4 b 2 J
, z p1,2J
2
(z 01J z 02J ) , a1J
b 2J
z 01J z 02J , a 2J
E
(z p1J z p 2J ) ,
z p1J z p 2J .
# a1 z p1 , b1
, " * >
" ?
a 1J r a 12J 4 a 2 J
,
2
z 01 .
z01=1, z02=z1, zp1=0.4+j0.6, zp2=0.4zj0.6,
%& . 2.15, > #>@ "' " 2- #:
b0=1, b1=0, b2=z1, a1=z 0.8, a2= 0.7211.
&% ++.
= # ' + " C; ( . 2.17)
# * # &$ ' * " + HJ(z), % & A:
52
H(z)
C
L
b 0 J b1J z 1
¦ H J ( z) , H J ( z)
1 a 1J z 1 a 2 J z 2
J 1
(2.50)
" # E
' & b1J a2J (2.50) & >.
&$ # * + &$ #&$ " + :
y( n )
C x (n ) L
¦ y J (n ) .
J 1
x(n)
y(n)
H1(z)
…
H2(z)
HL(z)
C
. 2.17. # % " " ,D
= # ' , >@ + * C;, " % & # # * '
+ , # * + * +- > *
.
= E # # * ' " + + * & HJ (2.50) # ' +< # " # * ' .
+ E
' & a1J a2J (2.50) #> >
@+> $ % < *, # # " C;.
E
' A
& % # # * ' " >
+ < M
C
z
z 0pii .
i 1
E
' & A, b0J, b1J % & " + % " E
' & # *
& C;, & "& > , + * " +>.
& ($ 1) "' # " ( . 2.18) & " & #
53
y( n )
b 0 x (n ) b1 x (n 1) b 2 x (n 2) [a1 y(n 1) a 2 y(n 2)] (2.51)
x(n)
y(n)
b0
z-1
z-1
b1
x(n-1)
-a1
z-1
y(n-1)
z-1
b2
x(n-2)
-a2
y(n-2)
. 2.18. " " ! ( 1)
A Z–1 $ " E "# % " # # # "' !#.
> " (2.51) E % " &$ * #
w (n ) b 0 x (n ) b1 x (n 1) b 2 x (n 2),
y(n ) w (n ) a1 y(n 1) a 2 y(n 2)],
& "#+
& > >
>
" *
"' .
* "'
" $ # & & E
, 5 $ #
5 % * ( b0 = 1 % * 4).
= * "' % & + 5 ( 4) ' *
% 4 ' % 1 .
($ 2) "'
" 2-
# # # * '
#>@ #:
H(z)
B( z )
A(z)
1 B(z )
A(z)
W (z) Y(z)
X(z) W (z)
H P (z) H H (z) ,
(2.52)
# W(z) – + ;
1
H P (z)
– # ' * " ;
1
2
1 a 1 z a 2 z
54
H H (z) b 0 b1 z 1 b 2 z 2 – # ' *
" .
A " ( . 2.19) 2 " +< E Z–1.
w(n)
x(n)
b0
y(n)
z-1
w(n-1)
-a1
b1
z-1
-a2
w(n-2)
b2
. 2.19. " " ! ! ( 2)
! " & # " & :
y(n ) b 0 w (n ) b1 w (n 1) b 2 w (n 2),
w (n ) x (n ) a1 w (n 1) a 2 w (n 2)],
(2.53)
" &$ " > *,
– * " *
. J # & # * &< " &$ * " *
, + >
> " .
D # , ($ 2)
"' " + &+ @& C;, # &$ &< 2. D# # +
& "'
>
#& @ " #+ &
+<* + > * " # . =
E +" * #
"'
' C;, " E
+
+<* $ # # +>, . . +> " .
?
. 2.20, 2.21 #& -$& (A) * "' C; # + # # &$ " + . = & ; > ' > .
55
' * J @ # +
@ # & " +
+ $ " * ' *. " ' > " & " (2.53) & ( # ) + * W2(J)=W1(J), W1(J)=W, & #< > #>@
< ' A.
?
D &$ E
' B0(J), B1(J), B2(J), A1(J), A2(J), W1(J), W2(J)
# L, B0(J), B1(J), B2(J), A1(J), A2(J)
J=1
# X=x(n)
W=M(J)X–A1(J)W1(J)–A2(J)W2(J);
Y=B0(J)W+B1(J)W1(J)+B2(J)W2(J);
W2(J)=W1(J); W1(J)=W; X=Y
J=J+1
J>L
0
1
& # y(n)=Y
. 2.20. < -
! ! D
( )
? A
. 2.20, 2.21 +" & & &
( #
&): B0(J), B1(J), B2(J), A1(J), A2(J) – # E
' 56
" + b0J, b1J, b2J, a1J, a2J X, Y, W, W1(J), W2(J) – # ' M(J) @x(n), y(n), wJ(n z 1), wJ(n z 2). C @+> E
<
$ #&$ " + .
?
D &$ E
B0(J), B1(J), A1(J), A2(J), C
' # L, B0(J), B1(J), A1(J), A2(J), C
# X=x(n)
J=1
Y=CX
W=M(J)X+A1(J)W1(J)+A2(J)W2(J);
Y=Y+B0(J)W+B1(J)W1(J);
W2(J)=W1(J); W1(J)=W;
J=J+1
J>L
0
1
& # y(n)=Y
. 2.21. < -
! ! ,D
( % )
= $ # *
"' " + " ' (2.51) " #
X=M(J)X;
57
Y=B0(J)X+B1(J)X1(J)+B2(J)X2(J)zA1(J)Y1(J)zA2(J)Y2(J);
X2(J)=X1(J); X1(J)=X; Y2(J)=Y1(J); X=Y.
#
X1(J), X2(J), Y1(J), Y2(J) > "#+
& xJ(nz1), xJ(nz2), yJ(nz1), yJ(nz2).
" # $ C; , " "' , # +% "@ ++ # " ( . 2.22). = #+&$ $ #& "> &$ # , & " , "& .
y(n)
x(n)
z-1
b1
-a1
z-1
b2
. 2.22. E" % -a2
" " ! " ' @ * & " > #>@ &:
X=M(J)X;
Y=X+W1(J); W1(J)=B1(J)XzA1(J)Y+W2(J); W2(J)=B2(J)zA2(J).
&$ +&$ * & +
&, #>@ # % + * , # %& &+ & $ . J & W1(J), W2(J) #
*
& " +
X1(J), X2(J), Y1(J), Y2(J) – # *.
" & % ' 4' ++' C;, +"& $', ,.
2.12. *// :* *(A9 &(*(*%&'4 :8)*
? + , & A, $ . 2.23.
* "'
?C; $ # & (N z 1) E
, N % * N $ # .
58
x(n)
z-1
h(0)
z-1
z-1
h(1)
z-1
…
h(2)
h(3)
h(N-1)
y(n)
. 2.23. " " " ! % E$
?
D &$
H(k), X(I)
E
' # N, H(k)
I=0
# S=x(n)
X(I)=S
k=0, Y=0
Y=Y+H(k)X(I)
k=k+1
0
k=N
0
I=I+1
I=N
1
I=0
1
& # y(n)=Y
. 2.24. < -
! ! G,D
! E$
59
-$ * "' ?C; # . 2.24. # «< &$» &$
Yy(n), H(k)h(m), X(I)x(nzm). ; + " " > ' >
Y=Y+H(k)X(I). = & X(I) # %& &+ & $ .
! &* G & * * "'
(N z 1) ' * % N ' * % %#&* &$ # .
2.13. (*(-)&/ :&9/ %))&/ 4*)(*%)
&(*(*%&0 :8)*
= # ' ?(z)
$ ?(jq)
?C; #> Z- " " ; + + * $ :
N 1
H(z)
¦
h (n ) z n , H( j Z)
N 1
¦
h (n ) e
jZn T#
(2.54)
n 0
n 0
= +< * " " HF & + & >
# # & +& $ , >@ +< G & *.
% * & > *
+
" % + * * ;HF ""#& (U).
* ;HF + * $ + : h(n)=h(Nz1zn). D >@ # > ?C;
""#& t"=z[(Nz1)/2]T#.
> ;HF: M(q)=zqT#(Nz1)/2
2.14. (*(*%&'( :8)*' % &(6&6 :A%))&6
4*)(*%)6
= # > ' >
> $ ?C;
(2.54) + * $ *, >@* > h(n)=h(Nz1zn) ( . 2.25), N % #
H(z)
­
®
°̄
( N 1 ) °
z 2 h ( N 1)
2
­
H( jZ) e
jZT# ( N 1) °
2
N 3
2
½
( n N21 ) º °
ª ( n N21 )
¦ h (n ) «z
z
»¼ ¾
¬
°¿
n 0
N 1
®h ( 2 ) °
¯
N 3
2
¦
n 0
60
>
(2.55)
½
@
°
2 h (n ) cos ZT# (n N 1) ¾
2
°
¿
(2.56)
h(n)
n
(N-1)/2
0 1 2 …
N-1
…
. 2.25. # " % G,D
= N h((Nz1)/2)
$ & % $ , $ * # " (N/2)z1.
" & % # HF #, ;HF + M(q)=–qT#(Nz1)/2 z *, ""#& t"=z[(Nz1)/2]T# – " &.
= # * '
(2.55) % +
?C;, >@> # +< ' *
% ( . 2.26).
x(n)
x(n-1)
z-1
z-1
h(0)
z-1
x(n-2)
…
z-1
z-1
z-1
…
z-1
z-1
h(1)
x(n-(N-1)/2)
h((N-3)/2)
h((N-1)/2)
y(n)
. 2.26. " " G,D " % ! " y(n ) h ( N 1) x (n N 1) 2
2
N 3
2
¦ h (m) >x (n m) x (n ( N 1) m)@
m 0
> % + # N.
A# +, N |H(j q#/2)|=0 M(q#/2)=0.
61
?C;, C;, % @ > #+ ( . 2.27)
< &.
x(n)
@-
…
h(N-1)
h(N-2)
z-1
h(N-3)
z-1
w(N-1)
…
h(1)
z-1
h(0)
y(n)
z-1
w(N-2)
w(1)
w(0)
. 2.27. E" % " " G,D E$
D @ * @ " * ' *:
Y=H(0)X+W(0); W(k)=H(k)X+W(k+1), k=0, 1…Nz1.
= & W(k), k= 0, 1…N, # %& &+ & $ .
2.15. *(*' *(D(&/ A- + )(* 9:*'4 %0&
> 1.
& '
* & y(n ) x (n ) b1 x (n 1) , b1=2.
D # + ' > & +&* #
­1, n 0, 1;
x (n ) ®
¯0, n ! 1.
U4
= &$ +&$ $ x(–1)=0 & #> # &.
y(0) x (0) b1 x (1) 1 2 0 1, y(1) x (1) b1 x (0) 1 2 1 3 ,
y ( 2)
x (2) b1 x (1)
# , 0 2 1 2 , y(3)
x (3) b1 x (2)
&$ #& & y(n )
> 2.
& '
0
* & y(n )
0 20
0.
nt3 & >.
x (n 2) .
D # + ' > & # x (n ) a n , a 1 .
D # + Z- " $ # &$ # .
U4
1. = &$ +&$ $ x (1) 0 , x (2) 0 & # > # .
62
, n 0, 1;
­0
! # &$ # * # y(n ) ® n 2
, n ! 1.
¯a
2. Z- " $ # * * (& >@* a 1 ).
f
f
X(z)
¦ x (k ) z k
f
k 0
k 0
3. Z- " &$ # k
¦ a z 1
¦ a k zk
1
1 a z 1
k 0
-
#,
+"
* Z^x (n m)` z m Z^x (n )`:
Y(z)
z 2 X(z)
z2
1
.
1 (a z 1 )
> 3.
* Z- " X(z)
1
.
1 5 z 6 z 2
1
D # + # # + x(n).
U4
1. D Z- " % &+ & X(z) z n 1 :
'
¦ Re s>X(z) z n 1 @ ¦
x (n )
>
zoz
n 1
lim (z z k ) X(z) z
x (n )
1
1 a z 1
>
z oa
N
X(z)
# X(z)
Ek
¦ 1 D
k 1
2. =
k z
" %
1
1 5z 6z 2
1
@
> @
z oa
n
lim z
an .
Z- " , , x (n )
1
N
¦ Ek
-
Dk n .
k 1
# *
+
& # > z=a.
n 1
z
lim (z a ) z a z
& *
* Z- " #
@
,
k
zk – > ' X(z).
'
# X(z)
&
# '
X(z)
1
,
1 5 z 6 z 2
1
> (zp1=2, zp2=3) &.
3 3 , x (n )
1
1 2z
1 3 z 1
63
(3) (2) n (3) (3) n .
> 4.
" & $ # *
&$ # * '
* &:
x (n ) ^1, 0, 1, 2`, y(n ) ^0, 1, 2, 1`.
D # + Z- " $ # &$ # , % # > ' > &.
U4
f
1. Z- " # & % X(z)
¦ x (k ) z k .
k 0
D#
# " & + & 4 # + * ( +& & >). A# + ,
X(z) 1 z 0 0 z 1 1 z 2 2 z 3 , Y(z) 1 z 1 2 z 2 1 z 3 .
2. = # ' & # < H(z)
Y(z)
X(z)
1z 1 2z 2 1z 3
11z 2 2z 3
1
2
z 1 1 2z2 1z 3 .
11z
2 z
> 5.
& '
* & y(n ) a y(n 1) x (n ) .
D # + # > ' > &.
U4
& +> # y(n ) o Y(z) , x (n ) o X(z) ,
& * y(n 1) o z 1 Y(z) .
=
Y(z) a z 1 Y(z) X(z) .
# # & &
>
@
Y(z) 1 a z 1 X(z)
= # ' & # < H(z)
Y(z)
X(z)
1
.
1 a z 1
> 6.
& '
* & y(n ) a y(n 1) b x (n ) .
D # + +> $ &.
U4
+ $ & # ' > ­1, n 0
& # &* + u 0 (n ) ®
.
!
0
,
n
0
¯
64
= &$
> #
h ( 0)
h (1)
h ( 2)
…
+&$ $ y(1) 0 & # &.
y(0) a y(1) b x (0) a 0 b 1 b ,
y(1) a y(0) b x (1) a b b 0 a b ,
y(2) a y(1) b x (2) a a b b 0 a 2 b ,
h (n ) a n b .
> 7.
= # ' '
* &
H(z) b 0 b1 z 1 b 2 z 2 b 3 z 3 .
D # + +> $ &, % + * $ (F F).
U4
+ $ & # ' > ­1, n 0
& # &* + x (n ) u 0 (n ) ®
.
!
0
,
n
0
¯
& # > # &, &* # y(n ) b 0 x (n ) b1 x (n 1) b 2 x (n 2) b 3 x (n 3) .
h (0) y(0) b 0 x (0) b 0 ,
h (1) y(1) b 0 x (1) b1 x (0) b 0 0 b1 1 b1 ,
h (2) y(2) b 0 x (2) b1 x (1) b 2 x (0) b 0 0 b1 0 b 2 1 b 2 ,
h (3) y(3) b 0 x (3) b1 x (2) b 2 x (1) b 3 x (0) b 3 ,
h (4) y(4) 0 .
# , n>3 & + * $ & >, # + , + F- .
" # # %, & + *
$ + & E
' .
> 8.
5
* & H(z) 1 z 1 .
= # ' '
1 z
D # + + * $ & (F F).
U4
+" " % # * ' &
H(z)
1 z 5
1 z 1
1 z 5 .
1 z 1 1 z 1
65
D Z- " 1
1 z 1
'
# -
­1, n t 0
u1 (n ) ®
.
¯0, n 0
+ $ # Z- " # * ' &. A * "# % h (n ) u1 (n ) u1 (n 5) .
& # > # &
h (0) u1 (0) u1 (5) 1, h (1) u1 (1) u1 (4) 1 ,
h (2) u1 (2) u1 (3) 1 , h (3) u1 (3) u1 (2) 1,
h (4) u1 (4) u1 (1) 1 , h (5) u1 (5) u1 (0) 1 1 0 .
& + * $ # + , + F- .
> 9.
= # ' '
n>4 & >, -
* & H(z)
1
.
1 a z 1
D # + + * $ & (F F).
U4
+ $ # Z- " # * ' &. # + $ h (n ) a n . = + "+ ' > + nof (|a|<1), + F- .
> 10.
U# # ' '
1. H(z)
1 z 1 ; 2. H ( z)
1 0.2 z 1
* &.
1 z 3
; 3. H(z)
1 0.6 z 1 0.25z 2
1
.
11.2z 1
D # + *
+ &.
U4
'
&$ % &+ +" #>@ *
* +*: > # * ' # %& $
* Z- # # ? & + & # *
# # * z p
1. ;' *
# > z p
.
66
# +
z p 1.
&, + $ > $ f.
0.2 . = + z p
0.2 1
2. ;' z p1,2
# > z p1,2
0.5 1 *
0.3 r j 0.4 . = +
.
# > z p
* .
3. ;' 1.2 . = + z p
1. 2 ! 1
2.16. &)*8&'( +*%'
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
K '
&$ # + *, + $ &, *
.
C" & , "' '
+ *< $ E .
A "+ %# & # "
, % .
Z- " # '
&$ . D# Z- " . D Z- " .
= Z- " '
&$ .
A & $& '
&$ + .
" % "' &$
+
# * * ?
# + $ '
+ ,
'
& + & "& > + F F?
@ & '
& + ?
* & > E
' & &$ '
&$ + ?
# Z- " # &$ # + *, & & * > + '
&$ + ?
# # ' + " >?
* # +- >
# * '
+ " ?
%> > '
+ * Z- > ' >
+ % +
* > ?
* *
"'
&$ " + #?
+ A?
* G & +&$ ' * & + $ # ?
67
3. E =
# "# +" & #>@ : [1].
3.1. ))%)(%( 4*)(*%) +0*(D&%) &)&/
i
N/2
xmax
x (n)
e(n)
'x k
4
3
2
1
0
1
2
3
4
-N/2
nT#
1T# 2T# 3T#
xmin
. 3.1. ! " + = " < + # " , "& < +> . K % # # E * < # " + " # , * N, " ',.
C $ < * " # (,) ( . 3.2) ',, " ,, >@ # # * , ,.
K % # < ' %# " * * ' M ('x k )
x k 0.5
³ (x x k ) p( x ) dx .
x k 0.5
68
(3.1)
p(x)
'xk
x
xk-0.5 xk xk+0.5
xm
. 3.2. + !H
" ! !
()
x k 0.5
³
D('x k )
( x x k ) 2 p( x ) dx .
(3.2)
x k 0.5
= +< * % # % +, " * # * (xk) – " > ,=,.
! #
M ('x k ) p( x k )
x k 0.5
³ ( x x k ) dx
x k 0.5
>
1 p( x ) ( x
x k ) 2 ( x k 0.5 x k ) 2
k
k
0
.
5
2
@
, # , V('xk)=0.
< D('x k ) p( x k )
x k 0.5
³
( x x k ) 2 dx
x k 0.5
>
@
1 p( x ) ( x
x k ) 3 ( x k 0 .5 x k ) 3 .
k
k
0
.
5
3
=
, ,k # D('x k )
1 p( x ) 'x 3 .
k
k
12
+
, " # (,k) 'xk $ %# * # 'xk, D('x k )
1 >p( x ) 'x @ 'x 2 .
k
k
k
12
69
< " * # " " * 0 ,m
1
12
D('x k )
N
¦ p(x k ) 'x 3k .
k 1
& 'xk ='xi = const (izk)
'x 2k
12
D('x k )
N
= +
¦ p( x k ) 'x k
N
¦ p ( x k ) 'x k .
k 1
'x 2k
.
12
1, D('x k )
k 1
! " , & % # # < # +< * # > " (,).
A # # < V('x k )
D('x k )
'x k
12
xm
N 12
.
# # < "# , % # + $ # + * :
N
xm
xm
'x k
V( 'x k ) 12
.
A# + , $ # +< < # & >@ + * .
3.2. 0*(D&%)8 *&(*&0 &)&/
+* *A&'4 A&4 (( *%+*(-((&/
" & , % + % " * " , # & " *, & &$ <&
# ' &$ '
* .
= E # " , # # " %*< $ &$ " * n x 'x k n x r 1 'x k . "+ "-" " $ " + " ## < + .
70
x
x
x
x
D # " < # & $ " # < (',):
#$ ±'xk;
;
#$ ±0.5'xk;
+ " A .
p('x)
p('x)
'x
–'xk
0
'x
+'xk
+'xk
0
)
)
p('x)
p('x)
'x
–0.5'xk
0
'T
+0.5'xk
–'Tk–T0
)
+'Tk+T0
0
!)
. 3.3. + !H # & $ " # < ('$)
U'* '* # #$
±'xk ( . 3.3,) &$ '
&$ " +&$
$ $, .
K + " < 'x max
=
r 'x k .
# < +
Jk
r
71
'x k
xm
.
A # " < M ('x ) 0 .
A # # < "+ " n x 'x k , . . # ( . 3.3,)
V('x )
2
'x k
³
< > < > -
p('x ) 'x 2 d'x
0
'x k
.
3
U'* '* # ( . 3.3,)
'
&$ " +&$ $ '?@
+4, &$ #$ # @ # * &, + &$ " * # " n x 'x k # nx 1 'xk # "+ " n x 'x k .
E + < +
'x max
=
r 'x k .
# < +
Jk
r
'x k
xm
.
A # " < M ('x )
'x k
.
2
A # # < "+ " n x 'x k , . . # ( . 3.3,)
V('x )
'x k
³
< > < > -
p('x ) 'x 2 d'x
0
'x k
.
3
U'* '* # ('x) #$ ±0.5'xk ( . 3.3, ) '
&$ $ '?@ +4 # #$ # $ $+?@+? + $*
$$, * 0.5'xk. + &$ " *
, " # " (n x 0.5) 'x k #
72
n x 0.5 'x k # "+ " n x 'x k . J % " # $ + +% " . E + < +
'x max
=
r
'x k
.
2
# < +
Jk
r
'x k
2 x m
.
A # " < M ('x ) 0 .
A # # < "+ " n x 'x k , . . # ( . 3.2, )
V('x )
2
0.5'x k
³
< > < > -
p('x ) 'x 2 d'x
0
'x k
.
12
;+%'* '* # ( . 3.2,) $ +?@ $+%
# ;0. ! +&* " # " # $ &$ &$ " # –
T0 0 0 +T0. E + < +
'T max rT0 .
=
# < +
Jk
r
T0
Tm
.
A # " < M ('T) 0 .
A # # < "+ " n x T0 , . . # ( . 3.3,)
V('T )
2
T0
³ p('T) 'T
2
d'T
T0
³
0
73
< > < > 2
0
1 (1 'T ) 'T 2 d'T
T0
T0
T0
6
-
.
3.3. 0*(D&%)8 ) &)&/ +* A(*(&
%*(-&(-*)&0 %*(-&(0 A&(&6 (&'
# " # > & " " * &, # # # & " " #&* % . J & #,
# $ # ' > & " &, # # # & " "#&* % . ? , < $& E # & " #+&$ &$ , # # &
" " #&* % . E * "# * < " &$ " * " * & #+ ' " .
U# # * " > " # # " & " % # * '
&$ , " >@ $, , #* >@ " # $ , ">@ $ " * *. $ $ " # "+
" &$ " *. D # #* >@
# " * " &$ " $
$ # " # + @+> '
&$ .
D # * "# * < " # # # " * "+
" &$ " *. A , ' * &# ' ' '
> # # " . = # +$+? $$? " +& & .
& < # # # " *# # # & " ,(t) %
;=ti+1 – ti (%# &# # $ # &$
/
" *) $ x x . ; – # + +
– " "& ,(t) # .
A # # " & ,(t) # $ # "+
" &$ " *:
T# / 2
x 1
T#
2
³ >N 'x k @ dt
N 'x k ,
T# / 2
;.
# N – '
A # # " ,(t) # :
74
T# / 2
/
x 1
T#
2
'x º
ª
³ «¬ N 'x k t T#k »¼ dt
T / 2
>N 'x k @
2
'x 2k
.
12
#
= < + # # # " # ;:
/
x x /
x J "
N 'x k >N 'x k @
2
'x 2
k
12
N 'x k
1 .
24 N 2
|
" # # &$
H < N " *
, ,(t) + $ +< $ #$ " ,,
N|
1
.
24 J "
"# J " 0.1% , N=7.
?*# $ # < , ,(t)
4 $ 0 , $ + +%.
/
! # # # " x T
/
x 2
ª 'x k º
³ «¬ N T »¼ dt
0
'.
1
T
# ! – # + + N 'x k
3
,
A # # " x $ x 1
T
N
¦
(k )
T
N
³
T
k 1
( k 1)
N
ª ( k 1) T ( k ) T
N
N
«
2
«
¬
2
º
2
'
N
x
ª
º
k
»
dt
» «¬ T »¼
¼
! # < + " ,(t) J "
/
x x /
x N 'x k
3
(1 1 2 ) .
8 N
# # # &$ $
N'x k
N'x k
(1 1 2 ) 3
3
8N
N'x k
3
12.
8 N
H < N " # # &$
" * , x(t) * $ 0 x
N
1 .
8 J "
75
"# J " 0.1% , N=12.
x(t) " $ + +', < +
" # # &$ " *
J "
/
x x /
x 0.4 .
N3
= J " 1% $ # , & N=10, J " 0.1% $ # , & N=50.
A# + , &$ J " * " # # &$ " * & # "&$ " " ,(t) .
D# "# " +& # " " , $ # +, < + J " +< ,
+ & . = E $ # + *
# "# " < %+
"
# " , #
* " ,.
% "+, < + " ' " ,(t) " + J
"
,(t) x x /
x /
0.01 ,
N2
#+ " 0.084 .
J
1.43
N
=
J " 1% $ # , & N=5, # , & N=23.
J "
0.1% $ -
3.4. +*(-((&( +0*(D&%) ) &)&/ % ()
%*&/ % --)&6 +0*(D&%)8F +*(*A&/
& & " " +&$
" *, &$ " > < . H
&$ " +$ " ## < + 'X ## , #
" . =
#
E $
$ * < " %& # #$ #.
& $ $, >, * 'x k < 'X ## . ! # " x #
& %# *,
" , x 'X ## # # (AD) * < 76
V6
#+ 'X 2
##
'x 2k
.
12
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82
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f
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84
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# = – ' .
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x ( t )
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k 1i 1
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91
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2) # + HF
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3) + "$ HF "# % ", #.
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* HF |Hd(jq)|, >
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(1–~1)
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A " C; " % " , ">>@ # "' #
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f
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5.3.1. = # * " " ; &* &*
+ - (;=)
# * ' * ?(s) * $ * H(j¡), # " "& # * ' * H(z) * $ * H(jq) ;:
;=
;
;=
s f (z)
H(s)
o
z f 1 (s )
m
H ( j :)
H(z)
93
;
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o
Z f 1 (:)
m
H( j Z) .
A "+ E # * s=f(z)
* z=f–1(s) ">@ ' >@ s=j¡ z=ejqT# " ¡=f(q), q=f–1(¡) '
+. A @+> E $ " * #> ;=,
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> # >
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x S- + s=+j \, <0, * "@> > *
;=, # % # %+ +
# # |z|<1, Z- "@> > *
;, . . *
;= # %
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. .
# $ #, %+ % + # # jZT
#
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&$ $ $ + .
J "* $".
5.3.2. * " # #>@ " :
(5.4)
s=f(z)=(2/T)[(1–z–1)/(1+z–1)].
K % % * < z–1=[(2–sT)/(2+sT)].
(5.5)
" * '# & $ # * " #, + S- % # >
% + Z- (# |z|=1)
Im[z]
Im[s]
Re[s]
Re[z]
. 5.2. " ! 94
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K # '
&$
+
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'
H(s)
'
+ H(z) H(s) |s (2/T)(1- z -1 )/(1 z -1 ) .
(5.6)
= E " # $ + & $ , * * + . D# E ", & $ '
+ # &, # + $ «
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0 <:< f, >@ * '
* + , &* @+>
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$ %# >@ + " $ (;=?H). #+*< +" #$ #@ " # # E " $
&* ;=. ? ', $+' " E &* + " %&* '
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= '# & $ # # $+%* , <* > $
< $ #
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( "-" * < %# '
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< $ #
+
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> E $ # +" # * #$ # '
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E $+ " # # & '
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+ #$ #@ $ "# % # >@ #G & .
C + " $ ( * :=1 #/), # >@ "#&
$ .
C + " $ ( * :=1 #/), # >@ "#&
$ .
;=?H
;=?H
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"
;, #
#G
.
;
;=?H
>@ *
& -
;
)
)
. 5.3. #" %
97
5.3.4. C !"- (#$!")
A " ;=?H > & >@* ' , # # + m, " * * s0i > spi # * ' "#& & ¡ = 1, ¡" # < ' ~1, ~2 (A, A"). A " & @+> ' +&$ +> &$ .
?
> "
;=?H +> #>
# > ' > H(s):
m1
H(s) C
(s s 0 i )
i 1
m
,
(5.8)
(s s pi )
i 1
# A – >@ * % +; m1 – &$ * (m1<m).
A# +, > ;=?H > @ &
- %& ( " # + * +>), & & .
A " ;=?H "> ' "# * # "
* HF @+> >@ $ >@ $
' *.
>@ $ ' * +"> &
# . +& ' !* ( +& ), H&< , # & – E –U (E + &), H&< .
= # & '
+
+ * ' * > &$ *, $ & $ & "# % .
+
# * ' * # & '
> &$ $ "# % , &
$ – +' ( &) E * . ; + & H&< E > & +'
.
! & &$ $ " ;=?H + * # * ' #& . 5.4.
&$ $ & +' $ "& >@ & * > ¡pi,
¡0i =;.
; + & # * ' * > < $ "$ # #
+ +<
" # "# "$ * $ .
98
. 5.4. < ! D#GC,
"+ "+ " % .
' ;=?H H(s)
C
n
#
,
(5.9)
(s s k )
>
j S 12 ( 2k 1)
2n
k 1
@
# s k e
, A – .
? # + # "# > " * :".
n
lg(A 2" 1)
.
2lg(: " )
(5.10)
% ^"'4 1.
' ;=?H H&< 1 H(s)
C
n
#
,
(s s k )
k 1
# s k
Vk j \ k , Vk
sh (M) sin( 2k 1 S) ,
2 n
99
+ n #
(5.11)
\k
J J 1
ch (M)
2 ,
ch (M) cos( 2k 1 S) sh (M)
2 n
,
J J 1
2 ,
1/ n
2 º
ª
, H – +'
+ .
J «1 1 H »
H
¬
¼
? # + H&< 1 # "# > " * :" +'
n
lg(g g 2 1)
lg(: " : 2" 1)
,g
A 2" 1
H2
.
(5.12)
% ^"'4 2 ('*).
' ;=?H H&< 2 ( ) #
n
(s sn k )
H(s) C kn 1
,
(5.13)
(s sp k )
k 1
# sp k
V k j \ k , sn k
j
:"
cos( 2 k 1 S)
2 n
sh (M) sin( 2k 1 S) E k
2 n
,
Dk
, Vk
: " D k
D 2k E 2k
, \k
ch (M) cos( 2k 1 S) sh (M)
2 n
,
1/ n
: " E k
D 2k E 2k
,
J J 1
2 ,
ª A A 2 1º
"
«¬ "
»¼ .
? # + H&< 2 % # "# > " * :" +'
, & % (5.12).
ch (M)
J J 1 J
2 ,
5.3.5. $% #$!" &*
+ - " $ (;=?H) " + - (;=) @+>
#>@ $ &$ " *:
s
D&E^-DE^: s o
( + " $ );
:u
:
D&E^-D ^: s o u ( + & $ );
s
s 2 :u :l
D&E^-D&: s o
( * + );
s(: u : l )
100
D&E^-DU: s o
s(: u : l )
s2 :u :l
( % &*
+ ).
:u – $ ", :l – % ".
= &* ;= " &* ; @+> * " (5.6).
;=?H % &+ " ;=?H * " (5.6). & > & " # ;:
1
z 1 D
sin(
>Zc Zu @
T)
2
;
>
1
Zc Z u @
1- D z
sin(
T)
2
>Zc Zu @
1
cos(
T)
z D
2
]&E^-] ^: z 1 o ,D ;
>Zc Zu @
1 D z 1
cos(
T)
2
z 2 2D k z 1 k 1
1
k 1
k 1 ,
]&E^-]&: z o
k 1 z 2 2D k z 1 1
k 1
k 1
>Z Z @
cos( u l T )
>Z Z @
Z
2
D
, k ctg( u l T ) tg ( c T) ;
>Z Z @
2
2
cos( u l T)
2
>Z Z @
z 2 2D z 1 1k
cos( u l T )
1
1 k , D
k 1
2
]&E^-]U: z o
,
>Zu Zl @
1 k z 2 2D z 1 1
cos(
T)
1 k
k 1
2
>Z Z @
Z
k tg ( u l T) tg ( 0 T ) .
2
2
]&E^-]E^: z
o
,D
>
>
@
@
> @
> @
Zu – $ ", Zl – % ", Z0 – ' +
=; C;, Z – " ;=?H, T – # # "' .
5.3.6. $ & C + '
*
+ : + – =; ( *); ' – ; & " – 50 ', 150 '; # 300 ' – 20 #; # "' – 1 '.
U4
1. C
+ & # n + " $ (;=?H) ' .
101
n
lg(A 2" 1)
,
2 lg(: " )
# :" – + "# % , " – E
' "# % .
= "# > $ " F2=150 ', "# % F"=300 '.
F
A# + , + "# % :"= F "
2
300
150
2.
E
' "# % "# # '$ # (20 #).
& " +&$ # '$ ("=10).
=
" # # + n
lg(10 2 1)
2lg(2)
= 3.31.
D +<> # n=4.
2. ;=?H 4 # &# #>@ " H(s)
1
,
n
(s s k )
>
j S 1 ( 2k 1)
k 1
@
V k j \ k – >
# s k e 2 2n
+ .
n + % " + 2 #
H(s)
1
1
s 2 2V1 s V12 \12 s 2 2V 2 s V 22 \ 22
Vk
V1
1
1
.
s 2 2V1 s 1 s 2 2V 2 s 1
( 2 k 1) º
cos(S ª 1 ).
2 n »¼
«¬ 2
( 211) º
cos(S ª 1 ) 0.383 , V 2
2 4 »¼
«¬ 2
( 2 2 1) º
cos(S ª 1 ) 0.924 .
2 4 »¼
«¬ 2
3. =
" # " ;=?H &* + (;=) "# ( # * + ).
= " # * # > ' > H(s)
s 2 :u :l
, # :u – $ ", :l – % so
s(: u : l )
". ( : u 2 S Fc 2 , : l 2 S Fc1 , ': : u : l ).
' % * " & :l $ # # + Z12
Zc12
2 F# tg (
:l
)
2 F#
2 F# tg (
S Fc1
)
F#
2
H(s)
k
2 103 tg (
S 50
103
) =50.41.
1
2
>Zc12:u @2 1
2Zc12:
2V Zc12:
2V
2
1 s 2 ':k s (1 ': u ) k ': u 1s s
':
': 2
102
.
4. & * " ;=
* + .
'
"#&*
s o (2F#)[(1–z–1)/(1+z–1)].
2 B B z 1 B z 2 B z 3 B z 4
0,k
1,k
2,k
3,k
4,k
A 0,k A1,k z 1 A 2,k z 2 A 3,k z 3 A 4,k z 4 .
H(z)
k 1
2
ª 2 F# 2 2 Vk 2 F# §
2 Zc12 Zc2 · ª 2 Vk Zc12 Zc2 º ª« Zc12 Zc2 º» »º
· C0 « §¨
1
»
¨
¸ «¬
k « © dB ¸
dB
2 F# dB
2
¹
¼ «¬ ( 2 F#) 2 dB2 »¼ »¼
dB
¬
©
¹
B
C0
A
2 ºº
2
ª
ª
« 4 § 2 F# · 2 §¨ 2 Vk 2 F# ¸· 2 ª« 2 Vk Zc12 Zc2 »º 4 « Zc12 Zc2 » » C0
k
« ¨© dB ¸¹
«
dB
2 F# dB
2
2 »»
©
¹
¬
¼
¬
¬ ( 2 F#) dB ¼ ¼
2 ºº
ª 2 F# 2
ª
· 2 ª 1 2 Zc12 Zc2 º 6 « Zc12 Zc2 » » C0
« 6 §
«
»
k
« ¨© dB ¸¹
«
2
2
2 »»
dB
¬
¬
¼
¬ ( 2 F#) dB ¼ ¼
2
ª
« 4 § 2 F# · « ¨© dB ¸¹
¬
2
ª 2 F# 2 § 2 Vk 2 F# · ª
2 Zc12 Zc2 º ª 2 Vk Zc12 Zc2 º ª« Zc12 Zc2 º» »º
· ¨
Ǥ
1
C0
¸ «
»
» «¬
k
« ¨© dB ¸¹ ©
«
dB
2 F# dB
2
2
2 »»
¹
¼
dB
¬
¬
¼
¬ ( 2 F#) dB ¼ ¼
0 k
A
A
A
1 k
2 k
3 k
4 k
k
B
1 k
0
B
2 k
2 C0 B
3 k
k
0
B
4 k
C0
1
A
k
0 k
1
ª Zc12 Zc2 2 º º
§ 2 Vk 2 F# ·
ª 2 Vk Zc12 Zc2 º
«
» » C0
2
4
¸
«
»
«
»» k
dB
2
F#
dB
2
2
©
¹
¬
¼
¬ ( 2 F#) dB ¼ ¼
2 ¨
# ¢ 1
B
0 1
A
0 1
1
0.06527802
A
1 1
B
1 1
2.92520672
0
A
B
2 1
0.13055603 B
3 1
3.66251757
2 1
A
0
B
4 1
2.33655434
3 1
0.06527802
A
4 1
0.65819494
¢ 2
B
0 2
A
0 2
1
5. 0.0525718
A
1 2
B
1 2
0
2.69104173
A
B
2 2
2 2
0.1051436
2.94961679
A
B
3 2
3 2
0
B
4 2
1.54652942
0.0525718
A
4 2
0.33543104
& %# &# #>@ " y
j k
4
§ 4
·
¨
¸
B y
A y
t k j t k1
t k jt k ¸
¨
t 1
©t 0
¹,
¦
¦
# yj,k – &$ # * k- , yj,k–1 – $ # * k- .
$ #&* $ # * + " yj,0.
103
5.4. &)*8&'( +*%'
1.
2.
3.
4.
5.
' + . D & $ + .
= F- +
# * " : # # ".
A & ' #
' * " .
#& ' ;=?H. & + - .
U + # * ' > ( @ * #$ #).
& & " " # * " .
104
6. !$ -! I$
= # "# +"
3, 5, 7, 13, 17, 18].
& #>@ : [2,
6.1. &)(A &(*(*%&'4 :8)* ()- (%'4 :&96
6.1.1. A " ?C; (F- + ) & "# * # "
* * $ + Hd(jq) & ""#& # & < ' ( . 6.1). D
"> + * $ + h(n)N * # & N, >@* E
' # * '
N 1
¦
H(z)
h (m) z m .
(6.1)
m 0
& , $ + $ "& * " * ; +, @+> " ; + % &+ *# + $ hd(n), "# * # "
* * $ :
h d (n )
T#
2 S
Z# / 2
³
H d ( j Z) e
jZn T#
dZ .
(6.2)
Z# / 2
D# + $ (6.2) #+ + > # > " * " :
n < 0 hd(n) v 0 – + % $ # "#* .
= E % &+ # +" + * $ ?C;.
? , # '
;?H * ± q#/2
­1, Zc d Z d Zc ;
H d ( j Z) ®
¯0, # # $ Z ;
h d ( m)
T#
2 S
Z
³
H d ( jZ)e
jZ m T#
dZ
Z
105
Z T# sin(Z mT# )
S
Z m T#
O sin(O m )
.
S Om
. 6.1. @" % %! DGC
+&$ $ # $ ; #& . 6.1.4.
= + + * $ (6.2)
" "&* F- + * $ *, " * "# *, % # hd(n) (N – 1)/2 " # n < 0 n N. = E $ + & # ; + E
' hd[n – (N –1)/2]:
H( j Z)
N 1
¦
h d [m ( N 1) / 2] e
jZ m T#
(6.3)
m 0
" , # ; + %# " ["", " >@ '
" & &$ ' *.
< '
# &$ ' * +> $ ?C; > # & + * $ hd[n – (N – 1)/2] @+> ' +&$ &$ ' * w(n) * # & N:
h (n ) h d [n ( N 1) / 2] w (n ) .
(6.4)
= E % > $+%+?
+? +? wR(n) = 1, n = 0,..N – 1.
= * " + * $ $ + H( j Z)
N 1
¦
h[m] e
jZm T#
,
m 0
# * * "# * * $ Hd(jq) * $ * (; +z " ) * ' W(jq):
106
H( j Z) W ( j Z) * H d ( j Z)
T#
2 S
Z# / 2
³ W( j T) H d [ j (Z T)] dT ,
Z# / 2
# * – , £ – W ( j Z)
N 1
¦
w[m] e
jZm T#
(6.5)
,
– $ *
m 0
' .
& " * * >
> . 6.2, # #
%>@ '
"# * * $ & # ; +.
H $ * ' . 6.2 &* <
* ¤q & , + &$ $ " +& #> " ~.max @#+> #
& . A * @ @ #$ ± q#/2 " + % * * $ * '
& @# & "# * * $ * Hd(jq).
. 6.2. < + G,D
" 107
"
#, $ # * $ + H(jq) # <
* * $ * ' : 'Z | 'Z , < ' (+' ) "# % ~1, ~2
"& &$ . J # * ' , # % +:
¤q;
x +> <
x +&* + &$ ~.max +>
@#+ # & ;
x +> # N.
! E # &. !, # & '
> +< * + &$ , +<> <
, +<>@> # &
* ' N. J G " +"&$ &$ ' *. C & &$ " $.
6.1.2. '% & *
. 6.1 #& +"& " ; &
&$ ' *: + *, + *, F, FE E.
" * <
& ¤q=Dq#/N, # D z
"& &* D- , + &$ ~.max >> % ' & " < ' * $ "# % ( +& +' * $ ) |~2max|, #, &
# '
;?H * " = /4 [14]. ! % < > " ;H.
; # " (==;, =U;, K=;) " &$ #&$ < + ' %
&+ +< ' " , 6 #.
! ' 6.1
# " ! = +
! +
F
FE E
¤q
2q#/N
4q#/N
4q#/N
4q#/N
6q#/N
108
~.max, #
z13,6
z27
z31
z41
z57
~2max, #
z21
z26
z44
z53
z74
C # + #& . 6.1, % "$ > * $ "# % " #+ &
* ' .
&*4 + – $+% z +> < +&* + &$ .
(6.6)
wR(n) = 1, n = 0,..N – 1.
H $ ( . 6.3,) # & % WR ( j Z) e
jZ N21 T#
sin( Z N2 T# )
sin( Z 12 T# )
.
(6.7)
& * '
> <
¤q = q#/N ¤ = 2 /N. = = 0 |WR(j )| = N.
;+% + * # $ +&$ &$ ' * # * N/2:
w T (n )
0 d n d N 1
­ 2 n ,
°
w R (n ) * w R (n ) ® N 21n
°̄2 N 1 ,
2
N 1 n d N 1
2
(6.8)
# +< <
# +< &$ .
H $ + * * '
# * $ + * * '
* # &:
WT ( j Z)
sin 2 (Z N4 T# )
sin 2 (Z 12 T# )
.
(6.9)
& > <
¤q=2q#/N ¤=4/N.
!""@ + HK & & % w H (n ) D (1 D) cos( 2Sn ) .
(6.10)
N 1
= =0.5 * + H, =0.54 –
* + HK ( . 6.3,).
+ &$ * ' FE "& & # $ % * ?C;.
H > $ * ' FE ( . 6.3, )
% # + * $ &$ $ +&$ &$ ' * ' +& q0 = 0 q0 = ±q#/N:
>
@
>
@
WH ( jZ) DWR ( jZ) 1D WR j(ZT# 2S ) 1D WR j(ZT# 2S ) . (6.11)
!
!
2
2
109
)
)
)
. 6.3. C "! % " ( ),
" AJ! () ( )
& * $ > <
¤q=q#/N ¤=2/N. = @#+ # & 0.04 % @# # * $ * ' .
+ K #
w B (n ) 0.42 0.5 cos( 2Sn ) 0.08 cos( 4Sn ) .
N 1
N 1
(6.12)
= > * ' * FE < *
&* ( 1.5 ") + &$ .
H $ * '
E > * ' * FE # % # # +&$ &$ 0.04WR[j(q±2q#/N)]. ¥
&$ E * *
' ¤q=q#/N ¤=2/N.
= " ?C; +"> % E & & ' \' <, + -H&< , , # . [17, 18], # &$ " &$ ' * *" .
6.1.3. + ' & * ,
# $ &$ ' *, $ ">@ $ < & " &$ ~.max
D
'f N
f#
'f N (D- f#
), &$
' * *" E -
& <
+
+ @+> K _,
$ #@ & % E * ' :
> @
2
w A (n ) I 0 (E 1 2 n / I 0 (E) ,
N 1
# I0(x) z ' #.
110
(6.13)
# E < # # #
" ' "# * * $ +< * # + "# ' .
*" ' (. 6.2) & E
&, & " > # "# "$ > "=|~2max| (#) *
$ H(jq), >@* #+&* ;?H, & +
+ " D- E
' & ¦ [2]:
D|
A " 7.95
,
14.36
A " ! 21 #; D 0.9222, A " 21 #;
­0,
°
E ®0.5842 (A " 21) 0.4 0.07886 (A " 21), °0.1102 (A 8.7),
"
¯
A " d 21 #
21 A " 50 #
A " t 50 #
! ' 6.2
$ D- J L
" " + A", #
25
30
35
40
45
50
55
60
¦
1.333
2.117
2.783
3.395
3.975
4.551
5.102
5.653
D
1.187
1.536
1.884
2.232
2.580
2.928
3.261
3.625
", #
65
70
75
80
85
90
95
100
¦
6.204
6.755
7.306
7.857
8.408
8.959
9.501
10.061
D
3.973
4.321
4.669
5.017
5.366
5.714
6.062
6.410
! ' 6.3
" " % " , "+ " [2]
A", #
30
40
50
60
1 ±~1max, #
±0.27
±0.086
±0.027
±0.0086
A", #
70
80
90
100
1 ±~1max, #
±0.0027
±0.00086
±0.00027
±0.000086
= & " " '& " > D # $ # &* # + N§Df#/¤f , &* " # %*< +< .
111
# # $ &$ ' *, '
#+&$
+
==;, =U;, K=; "$ * $ "# % % &+ +< " , 6 #.
6.1.4. - ' % '% / +&$ $ ; " > @ & " ; + $ # "
&$ &$ $ HF Hd(jq).
% E^, " &<, +
$ # & % Oc
; h d (n )
S
h d ( 0)
O c sin(O c n )
, n=r1, r2, …
S
O c n
(6.14)
% $$+?@ % (=;) &$ # # $ #:
nz0; H d ( j Z) 1 Z d Z# / 2 . (6.15)
+& $ ; ^, & ( ),
U ( % ) V& ( ) &+ & %&
" +& $ '
E^
&:
H d ( j Z) ;H H d ( j Z) =; H d ( j Z) ;?H ,
(6.16)
y(n)=x(n); hd(0)=1; hd(n)=0 H d ( j Z) =;
H d ( j Z) C;
H d ( j Z) ;?H 2 H d ( j Z) ;?H1 ,
(6.17)
H d ( j Z) =; H d ( j Z) ;?H 2 H d ( j Z) ;?H1 , (6.18)
# Hd(jq);?H, Hd(jq);?H1
Hd(jq);?H2 – & $ #+&$ ;?H " c, c1, c2, (c2> c1), >@ " ;H, =; C;.
! % "+ # # +&$ $ , " # " + >@ & % :
h d (0) ;H
h d (0) =;
1
Oc
, h d (n ) ;H
S
O c sin(O c n )
, n=r1, r2, … (6.19)
S
O c n
O c 2 O c1
, h d (n ) =;
S
S
h d (0) C; 1 O c 2 O c1
, h d (n ) C;
S
S
sin(O c 2 n ) sin(O c1 n )
,
S n
S n
sin(O c1 n ) sin(O c 2 n )
.
S n
S n
(6.20)
(6.21)
& " $ # < # K=;.
112
6.1.5. ! '% & *
` 1. = "# " > "$ * $ "# % " @+> . 6.1 & * ' , >@* > |~2max| ", #, + " <
& , . . D.
=
+" * ' *" . 6.2 $ # >@ "# "$ > " & # * * ' ¦ D.
= E % & +, " "$ " # HF " + , &$ # & * ' N % "+ +<, +<
' " ~2max. H % HF + (=;, C;, K=;),
+< "$ # # * * % * ' . J % HF .
` 2. & * * '
"# * $ # *
& * $ + 'f f " f min
-
%& < ¤f=¤f =Df#/N $ # $ # # * '
# > # ' + * $ + : N t D f # / 'f , # D – E
, " @ * * ' (D- ), . . 6.1, 6.2.
U N %*< ' , &
.
` 3. A @+> " ; +
h d ( m)
T#
2 S
Z# / 2
³
H d ( j Z) e
jZ m T#
dZ
Z# / 2
#&$ &< $ & % *
@ + $ hd(m z (N z 1)/2), m=0…Nz1,
& -
>@ "# * * $ Hd(jq).
= E " "# * * $ +"> $ & " f , @& "# % $ # * &
+ ¤f
J " * & # # "& ' $ # & + "# % ( . 6.3). ? , # =;
f 1 | f 1 'f / 2 ; f 2 | f 2 'f / 2 .
113
` 4. ?$ # + $ + @ * (Nz1)/2 + * $ hd(m):
h (m) h d [m ( N 1) / 2] w (m), m 0,1,..., N - 1 .
` 5. C & HF + H( j Z)
N 1
¦ h[m] e
jZm T#
m 0
$ #& #& * $ A "$ > "# % A".
` 6. ! #&* # $ #&$
&$ #&$ ( '), $ # > " &$ " f1 , f2
# & + N
& >.
` 7. ?$ # + $ # " # + " *
+ * $ h(m) ( E
' + , " A), * HF @ # "#& .
` 8. & "' ?C; ( A =;)
<> >@ "# "' .
A# +, # &$ ' * > * + ;HF ""#& + # * * * E # + * $ h(m)=h(Nz1zm) (. . 2.13, 2.14).
6.2. &)(A &(*(*%&'4 :8)* ()- %))&6 '*
6.2.1. # * & + $ + "# * *
h(n)N $ # # "'
& # $ Hd(jq)
" ; + (D=;).
"' * $ Hd(jq) @ 0 … q# $ # & &$ " * & q # &: qk =¤qk, # k=0, 1, …, N z 1; ¤q=q#/N z < # "' ; k z * & ; N z # "' . C % & % jZ T
e k # * Z- # " . 6.4.
114
" * N
. 6.4. CA ` ¤q & " ¤qu¤q /(L+1), # L z '& , L = 0, 1, 2, …; ¤q z $ #
+ .
"+ # "
$ + (HF) H d ( j Zk ) H d ( j Z) Z Z ( . 6.5). ! "#k
$ " " + & ""#& , # ; "&
HF # "
$ %# # $ # "
* HF.
"' * $ . 6.5 & < ¤q=¤q /2 (L=1).
. 6.5. E CA ! % HF " , & 1 (Hd(jqk)=1),
"# % z > (Hd(jqk)=0)
$ # * – 115
& % & + & ( " &) " Hd(jqk)=H1=var, &$ " ' "# *
* $ .
HF Hd(jqk) % + > +>
$ hp(m), #> @+> # " ; + (D=;), # "' @ & % # + * $ hd(m), >@* "# * ( & *) * $ Hd(jq):
h d ( m)
Z#
T#
jZ m T#
³ H d ( j Z) e
dZ .
2 S
0
& "&: Z o Zk ; dZ o 'Z Z# / N; ³ o
+> $ h p ( m)
1
N
N 1
¦
N -1
¦
, k 0
hp(m):
H d ( jZ k ) e
jZ k m T#
k 0
1
N
N 1
¦
H d ( jZ k ) e
jZ k ( m i N )T#
k 0
# i = 0, ±1, ±2, ±… .
# «p» ", E + $ # * # Np = N, . . # "'
* # "' * ( . 6.6).
. 6.6. @" % , "+ ECA
+ * $ " # * & ?C; & # # + * $ hp(m), # &* (Nz1)/2 (# " * " )
&* + * *
' * (# F- + ) ( . 6.7):
h (m) h p (m N 1), m 0,1,...N - 1
2
= + * $ h(m) $ # $ + H(jq), >@ "#>:
116
H( jZ)
N 1
¦ h ( m) e
m 0
1
N
N 1
¦
jZ m T#
H d ( jZ k ) e
1
N
N 1
¦
H d ( jZ k )
m 0
N 1
jZ k ( N21 )T#
N 1 jZ m N 1 T
k
# jZ m T#
2
¦e
m 0
¦e
m 0
e
j(Z Z k ) m T#
m 0
( ZZ )
N 1
sin 2 k N T#
jZ( N21 )T# 1
e
H d ( jZ k )
( ZZ )
N
sin 2 k T#
m 0
>
¦
>
@
@
(
& # +" & % # & * ).
. 6.7. @" % G,D,
! jZ( N21 )T#
E & %
% + e
N
1
T# , + : M(Z) Z
2
# ;HF
* # -
+ * $ .
HF
+ $ q=qk: H(qk)=Hd(qk) # & & HF, $ qvqk H(q)vHd(q) z "# * < ' .
$$ '" * , $,* $ L
$ " * Hi. (i=1,2,…,L), #>@ $ > ' > # *.
C" & " L > #>@ &
" + &$ :
L = 0: ~2$ § z20 #;
L = 1: ~2$ § z40 #;
L = 2: ~2$ § z50 z 60 #;
L = 3: ~2$ § z80 z 100 #.
C+ # * & % "
+ ?C; +& "$ "# % # (90z120) #.
! " , "' + ">
& L z
&
$ # * $ +&$ "117
* Hi. , " >@ $ < ' . D # ,
+ &$ &
@ % '# "' . D # E " JK # * .
6.2.2. ! / '
` 1. = " > "# "$ "# % " & + &$ L * $ $ # * . ? , " u 40 #, L = 1.
H % HF + , +< "$ # " L.
` 2. " L "# * $ # * &
'f f " f $ # < # "' * $ : 'f
'f L 1
# "'
:N
f#
'f
L 1
f#
.
'f =
N %*< ' , & .
` 3. " "#> > $ "+ HF Hd(jqk), k = 0, 1,
Hd(jq) < ¤f,
…, N z 1.
D # k # &$, &$
+ &$ &$ &
.
U# +& " Hi. " &$ &$ &
%# * $ # * , , * *
' HF %# & " "# % .
` 4. C & > $ ?(jq) $ # &$ $ # " Hi. , "#& .
? , # ;?H
L = 1, N = 33 " H1 =0.3904, ~2max= z40 #;
L = 2, N = 65 H1 = 0.588, H2 = 0.1065, ~2max < z60 #.
` 5. C & +> $ ?C; * $ :
h (n )
H d ( 0) 1
N
N
¦ 2 H d ( jZk ) cos> n N21 Zk T# @,
KB
(6.22)
k 0
n = 0, 1, 2, …, N z 1, KB= (N z 1)/2 N KB= (N/2) z 1 – .
` 6. & "' ?C; (A =;).
118
6.3. &)*8&'( +*%'
1.
2.
3.
4.
5.
K #& F- F- +
(
' ).
= ' & F- +
# &$ ' *.
A * + * * ' : HF, + $ .
F #+&$ ;.
= ' & F- +
# * & .
119
7. $ #$ G
!$ ! I
7.1. *)(* %(&&0 %&)(A 9:*'4 :8)*
H & +& #& " ; "> JK @+> '# # * ' "#&$
&$ $ + #&
"'
< ' . = E &
$ + + " +>
. D &
' F F- +
> # # < (AD) < &< % ( &* *).
' "' AD # &% M
E
¦ > H( j Zi ) H d ( j Zi ) @
2
,
(7.1)
i 1
# H d ( j Zi ) , H( j Zi ) – "# >@ & $ + , & & # % ' + .
qi. J ' * + E
K &* * "> "' % +&$ " * " < ' < :
E(Z)
W (Z) H( j Z) H d ( j Z) ,
(7.2)
# W(q) – % + ' .
= +&$ " * E
' + * '
@ # +< $ # ,
* , * * "'
(
; -=E # F- + ) * "& C"
(#
+
&< * ' * F F- ). $ > E & +> & &, , K " +&$ > H&< F+ , +& & " ; FDAS2K, DFDP, Signal & MatLAB # .
120
" F F- +
+"> %
#& # #
, >@ & "' .
= " ; & # & > "+&
* ;HF F- +
*
F- + .
A "
& & # +& ; >
+<> ( # # > +>) < +
+ +< * ' "# #
# "# * (# *) < ' .
F- +
% $ + * &< *
' * > # + * $ N % # + "#& # < ' @+> E
*
&, # * [13].
7.2. &)*8&'( +*%'
1.
2.
3.
= '
&$ +
& # : ' , # +& &, & & .
A
+ * # # < (AD).
A
+ * < &< % .
121
8. "$
"G !I
= # "# +" & #>@ : [2,
3, 5, 7, 19, 20, 21].
8.1. +*(-((&( %6%) # !
" ; + (=;), > . 8.1,
& $' $" +% ( ) X( j Z) # * # + x(n) * # & N1,
& & # &$ @ $ $ qk= k ¤q:
=; N >x (n )@ X( j Z) Z Z
k
N1 1
¦
x (n ) e
jZ k n T#
,
(8.1)
n 0
# ¤q=q#/N – < # "' ; N – & &$
@ &$ & =; {0 z q#},
N1; k = 0, 1… N–1 – * & .
. 8.1. E !
& < # "' # " % +>
x(n) & X( j Z) =;.
# "
@ @+> " & (D=;). =; (8.1), D=; % &+ # "' & " ; +:
122
x (n )
T#
2 S
Z#
³
X( j Z) e
jZn T#
dZ .
0
+" "& dq q#/N; ; q q, $ # D=; N >X( j Zk )@ x p (n )
A xp(n) ±1, ..
1
N
# N 1
¦ X ( j Zk ) e
jZ k n T#
.
(8.2)
k 0
# N: x p (n )
" x(n) < x p (n )
x p (n i N) , i = 0,
¦ x (n i N) .
i
& N u N1 xp(n) = x(n), n = 0, 1.. N – 1, . . xp(n) 0…N–1 # $ #& x(n), # &
(N – N1) & # # % " # E ( . 8.2). D=;, & 0…N–1, # x(n) =;.
. 8.2. ! , "+ E#D N M N1
& N < N1 (¤q = q#/N > q# /N1) & # "
&$ # N # + * x(n) ( % * ), xp(n) v x(n) n = 0.. N1z1
( . 8.3). J > " % + # "
.
A < N N1 # & < # "' ¤q u q#/N1, % % * ":
$ * % "'% $ ' '", ' '4+'
4 $ wF.
123
& =; N, &<>@ # # + N1 (# > E (N–N1) & ), E '
, # "
+& # & < ¤q=q#/N1. x(n) & +" # &< "< =;.
. 8.3. ! , "+ E#D N<N1
! " , N- =; # "
* # N $ # * # + x(n) *
# & N1 u N.
=; # % " ; + # * # , & N, >@* *&*
# + xp(n) .
= " =; z D=; (8.1), (8.2) # > #
' # * & qk, * & k:
=; N >x (n )@ X(k )
N 1
¦ x (n ) e
j 2NS k n
, k = 0, 1… N – 1.
(8.3)
n 0
D=; N >X(k )@ x (n )
1
N
N 1
¦
X(k ) e
j 2NS k n
, n = 0, 1… N – 1. (8.4)
k 0
& D=;
=; N2 ' * % N(Nz1) ' * % &$ .
D " +"> # &* & +&* , &* $ # * " " :
D=; N >X(k )@
^
>
@`
1 =; X* ( k ) * ,
N
N
# * z ' % .
124
(8.5)
* &
=; # % * , & " ; + ( * +, # # + ), # +> *.
? % * # '
*
+ '
"+ =;
# &$ # + *. # &$ # + * " > > ($+?) *+? .
+ # # # $ # + * x1(n), x2(n) # N:
y(n ) x1 (n ) * x 2 (n )
N 1
N 1
m 0
m 0
¦ x1 (m)x 2 (n m) ¦ x1 (n m) x 2 (n ) . (8.6)
" , # + *
* % $ *, . . & " ; +
# $ # + * " # > " * ;+ E $ # + *: Y(k) X1 (k) X 2 (k) ( ).
& D=;, % @+> =; & + >
# $ # + *:
y(n ) D=; N ^=; N [ x1 (k )] =; N [ x 2 (k )]` .
(8.7)
C* # # &$ # + * x1(n) # * N1 x2(n) # * N2:
y(n ) x1 (n ) * x 2 (n )
N 1 1
N 2 1
m 0
m 0
¦ x1 (m) x 2 (n m) ¦ x1 (n m) x 2 (n ) . (8.8)
A * * y(n) # N=N1+N2–1. H &
+ # , =; # + * x1(n) x2(n) $ # & + # N,
>@ # # + y(n), # & < # "' ¤q=q#/N.
= E # + x1(n) x2(n) # > N01, N02
& : N01=NzN1, N02=NzN2, * ' > $ # "
.
A y(n) #& * % % &+
# @+> D=; " # N- &$ =; & &$ # + * x1(n), x2(n):
y(n ) D=; N ^=; N [ x1 (k )] =; N [ x 2 (k )]` .
(8.9)
& % (8.9) # & * *
&$ # + * * . =
125
+" &$ # & " ; + "& > % & * . D # , =; * * # + * * # &
x1(n), x2(n) E =; * # + *,
&$ # "'
$ # N=N1+N2z1.
8.2. !8)*9/ %0& & %&( # !
A * =; &$ # + * +">
# "' F- +
* * .
A &$ # + # # * *
* (A) $ # * # + x(n) ( # * # & N1) * + * $ * h(n) # * N2:
N 2 1
y( n )
¦ h (m) x (n m) , n=0,1, …N–1; N=N1+N2.
(8.10)
m 0
= & A
* "> & '
& + & (?C; A).
=
* A % &+ & (8.11):
y(n ) D=; N >H(k ) X(k )@, n=0,1, …N–1; N=N1+N2.
(8.11)
# "& > * % # + * * # & &. D # * $ * . 8.4.
E =; + * $ h(n)
H ( j Zk )
N 2 1
¦
h (m) e
jZ k m T#
m 0
Y ( jZ k )
X ( jZ k )
# "
* * $ + & $ #(HF), X( j Zk ) , Y( j Zk ) – # "
*
&$ # * # + *.
x(n)N
x(n)N1
X(jZk) Y(jZk)
=;N
[x(n)]
+N01
h(n)N
h(n)N2
+N02
D=;N
[Y(jZk)]
y(n)N
H(jZk)
=;N
[h(n)]
. 8.4. " " G,D E#D
126
> #>@ ' :
x " N1 $ # * # + x(n);
x & N- &$ =; # + * x(n) h(n);
x % N &$ & =; $ # * # + HF + " N- * # + Y( j Zk ) H( j Zk ) X( j Zk ) ;
x & N- D=; # + Y( j Zk ) ,
"+ > N &$ # * # + y(n).
! " , # & &$ # $ # & & $ # , " <& "# *
* $ *
+ . ; + ' @ # + $ # " * >, * $ #
>.
K % & + "'% ' $+%* ,, $ * * , H( j Zk ) .
D +> ""#& &# &$ # , & > +
* $ # * # + . "
E # " +, ""#& * " "& " "#+
&* $ .
"'
+ $ # + # " &$
# + * x(n), X( j Zk ) , Y( j Zk ) , y(n)
E
' H( j Zk ) # > N. D > % K 4 [2 N 2 N]
' * % K % 4 [( N 1) N] ' * % @ &$ . # &$ # E ' * K (1) 4 (2 N 1) K % (1) 4 ( N 1) .
= G & *
+ =; ?C;
*
* –
+ A (#
# & N2 ' % ).
D# E + @ " +" # & =; D=; & " ; + (=;). !, & =; > 2 >
+ % ' *
K 2 N log 2 ( N) ' * % % @ &$ . D@ # # ' * # ?C; =; E 127
K 4 N >log 2 ( N) 1@, K %
K (1)
4 >log 2 ( N) 1@, K % (1)
4 N log 2 ( N) ,
4 log 2 ( N) .
=
N = 1024 K (1) 44 , K % (1) 40 .
?C; A ' * " # & + * $ N2 N2 = N/2 K (1) K % (1) 512 .
! " , "' ?C; =; +< G ' *. = * '
+" # $ " &$ =; E + # * "'
"& @ &<. = G & * '
&
+ & =; & & '
&
+ ( G ).
8.3. +()*8&'6 &A %0&: A-, ()-', +*()*'
A +&* " ">
" %
& +& >@ ' " $
+&$ $ – #&, "&, @ , + * @ # .
"#, <& # + ", :
x % ;
' ;
x " < x % #&$;
&$ " ;
x &# ' G ( # &$, +&$
x #
# $ $ );
x " " ( , " % *) # .
*&$ @+> + " < @ "# & &&$ # * $ ( ' &$) "*.
A +&* " # &$ # $ ( &$) * # + "& > %
" [19].
D & # + " > %' ( #& "), "%' ( & =;), ( $ #* *&$ ' [20]), @ , +"@
@ + ", # + + , # + + , + + ( " ).
128
x
x
x
x
"
";
># " (<
) T N T# >@ # N & * "' ;
" 'f , &<>@ # # &$ * & ±f#/2;
" < , ' + >@ " # $ # $
": 'f p 1/T
" <&$ ( "#&$) &$ >@ $ .
A +&* " <
# $ %&$ > $.
D + " =;
"& ' * "' * # &,
. . ># . = E , "
# E > # # % * "' . ¥
"
=; & E &$ & +&$ =;.
8.4. +()*8&'6 &A %0& & %&( # !
"
, +">@ $ =;, % " , # . 8.5. D " " & ' " – " < & =;. &$ # =; $ # *
@ * # # + x(n), * * ' * w(n) * # & N:
~
=; N >~
x (n )@ X(k )
N 1
¦ x ( n ) w ( n )e
jZ k n T#
n 0
N 1
j
¦ ~x (n )e
2 S
N
k n
,(8.12)
n 0
k=0,1, …N–1.
U#+ ~
x (n ) x (n ) w (n ) – " $ # # + + =;; qk=kq#/N fk=kf#/N – & ", "& &
% " &: 1 < # "' * f#/N. "
N "&$ 1 (f#/N) " ' +& qk (fk), E " k=0,1,…N–1 > , ~
~
* & =; X( j Zk ) X(k ) . ' # , " ># $ # * , # * ># .
# " T=NT# 129
…
=;N
[x(n)]
~
x ( N 1)
…
~
x (n )
x(n)
~
X ( j 0)
~
X ( j 1)
~
x ( 0)
~
x (1)
~
X ( j ( N 1))
w(n)N
. 8.5. " " E#D
% > " < >
* *, E & =; # "
* * " X(jq) * $ * ( ) * ' W(jq):
~
X(j Zk ) X(j Z) * W ( j Z) Z Z , # * – , . . k
# % > ( # >) $4% . D
# # + , %>@ "+& + ".
= ' +&$ &$ ' * " # + + &"& & E " &
$.
+*< &$ #&$ #&$ =; @ " &$ ' &$ @+> =; +&$
$ , " @ $ # " &$ .
$, xp(n) # NT# ' >
#& A m (Zk )
"& M(Zk ) * kf#/N $
# " # @ >A m (Zk )@2 / 2 .
', * % x(n)
(
# $) ' >:
x +> + X(j Z) " +> [/'], #>
# | X(j Z) | M(Z) , . . #&
" &
& > $ " q=qk $ =;;
x E * +> + E Sx(q)
2
( X(j Z) ) " +> [2/'], "& >@> # E % & > # &$
$ qk.
+*', x(n) ' > +> +
@ Px(q) " +> [2/'], >@> < + * E *&
130
&, . . & * E * % & > # &$ $ qk.
, ', *&$ x(n), y(n) @+> =; " > $ " > +> + @ Pxy(q).
=
"'
&$ + "
" &$ % " + <
"+ " $ " [20].
=; # & % X ( j Zk )
N 1
¦ x (n ) e
jZ k n T#
.
n 0
@ $ xp(n) NT# kf#/N, #>@ =;,
#& #> A m (Zk ) 2 X( j Zk ) ,
"& – M(Zk ) arctg[X Im ( j Zk ) / X Re ( j Zk )] ,
# N
2
# @ 2 1 X( j Zk ) .
N
* # + NT#
& " $ # #&, "& @ k-* * & , + + $ qk # T#X(jqk). +& $ "& =; < :
Sx(k)=|T#X(jqk)|2 z + + E qk;
Px(k) =(T#/N)|X(j qk)|2 z + + @ qk;
Sx
Px
1
N T#
1
N T#
N 1
¦ S x (k ) – E ;
¦ Px (k ) – # @ + .
k 0
N 1
k 0
K %
+, + <
& E @ " * " & % T# # =; 1/T# # D=;, "& # - & # ; + (C;) [20].
8.5. +*(-((&( %%:9/ 0*) " !
& & " ; + (=;) – E &
& & =;, >@ * > =; & +> "& +. D & & # %& 1965 #
131
' !+>
" & DA
* . & & > *, $* K$' e
j 2NS k n
kn
x WN
kn
WN
:
( N k )n
WN
( N n )k
WN
;
( k l N )( n m N )
kn
WN
# , & # x $ WN
& * "' N ( =;).
pkn
A # * E WN
kn
WN
/ p -
# N/p, # p – '& , & # N.
+" #&$ *
$ =; >
+< >@ $ & =; ' *.
D@ * ' =; ">
" =; $ # * # + =; # # + * +<* # &,
+ # +
" % * ( * > =;), " &
& =; $ # * # + .
C" " % # + *
* * . " E " > & $ $ & $ $ .
=;, =; % & + + # N, >@ ' * m:
N=mL, # L – E E % : L=logmN.
+"& =; m= 2, 4,
8, @ > =; $ ? 2.
"
[7], . 8.8, @+> =; & %
=; (D=;).
8.6. 0*) " ! + %&&F 2 % +**(K&( + *((&
=+ "# # + + x(n) * # & N,
n=0,1,…N–1. ?% * =;:
N 1
X(k )
¦ x (n ) e
j 2NS k n
n 0
N 1
¦ x(n ) WNk n ,
(8.13)
n 0
# k = 0, 1,…N–1 ( =;) +& G & *. C< E * "#
# =; $ # #>@ " .
$ #> # + + x(n)N # * N " + 2 # # + # * N/2 – > ( >>@> & x(n)
&
# n: x1(n)=x(2n)
>: x2(n)=x(2n+1),
132
n = 0,1,…(N/2)–1. J ( . 8.6).
. 8.6. @ + !
%
> D " $ =; X1(k)N/2 X2(k)N/2. & " =; $ # * # + x(n)N " =; # # + * x1(n)N/2, x2(n)N/2:
N / 2 1
X(k )
¦
x1 (n )e
j 2 S k n
N/2
N / 2 1
¦
x 2 ( n )e
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e
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n 0
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(8.14)
k
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,
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J & N/2 &$ & =;.
> &$ & X(k) # k=(N/2), …(N–1)
*# * # :
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X(k N ) X1 (k ) X 2 (k ) WN
2
k = 0, 1…(N/2 – 1).
& % (8.14), (8.15)
( ' > G# ):
k
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, (8.15)
#> "+? $?
&
k
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,
k
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.
2
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k
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>> # % % – & .
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? " ' > % ( $ * &$ #)
& ( % * &$ #), % > k
>@ * % + WN
.
133
X1(jk)
X(jk)
k
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X(j(k+N/2))
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. 8.8 # N=8.
. 8.8. ! % ! ?#D ! J D' &* G & * #&
' * % :
# =; K.=;=N2;
# =; K.=;=2(N/2)2+N/2=N2/2+N/2.
# ,
"+ # % G & * +< 2 ".
+< %#> " # + * x1(n) x2(n) % " + @ # # # + # +<* # &: x11(n),
x12(n) x21(n), x22(n) (> >) + &<
#& '
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X L ( 0)
x L (0) x L (1) W20 ,
X L (1) x L (0) x L (1) W20 .
"+ &* # N=8.
134
=;, "&* . 8.9
X(j0)
xp(0)=x(0)
xp(1)=x(4)
xp(2)=x(2)
xp(3)=x(6)
xp(4)=x(1)
X(j1)
0
W2
0
W2
0
X(j2)
W4
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0
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1
xp(5)=x(5)
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0
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W8
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1
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%# " L E
& –
G# =; & > N/2 " &$ ' *, @ * G
& * # &$ ' * % % –
& :
K .=;
N L
2
N log N
2
2
,
K %.=; N L N log 2 N .
(8.17)
H ' * @' 4 " +< #
% 2 " +< # % – & . & &< =;
+ =; ' * % K .=; / K .=; 2 N / log 2 N .
N =210 = 1024 K .=; 5120 , K .=; | 10 6 & &< 204.8.
&#& . 8.9 " & > * * + * . ! & & >
E , " % @ $, #>@ @> > +> +
G, 2N # N
&$ ( $ + * * ). = E +"& * $ , N $ # x(n) ( @ ) @? N
& & & =; X(k).
D +> =; % &* &* # $ # ,
&* & " & &
# # + (n = 0, 4, 2, 6, 1, 5, 3, 7 # N = 8). ! * !, 135
# # "& > -'. J # $ # # + * $ # * # + # & *.
E & # + x(n)
# > L- " # #
#, #& E & >
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, >@> *
# + x(p).
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# * > #
&* #
n(2)=100, #
- &* ( &*) # n# . .=001 # &* p=1 * # + x(p).
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. 8.10. < -
! ( )
136
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. 8.11. < -
! ! ?#D
+ 2
137
A " * " ' & =; % " + %&$ ' ( # $
% ):
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& =; m = 1, 2, …2i–1.
x " * '
U >@ $ % * # " * ' i E #> @& & % Wk
N/2
L i
>
@
k=0,1,… N/(2 L -i 1 ) 1 ).
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D >:
x ( G ) +"&$ &$;
x
# N & * # + ( )
x(n) ( X(n));
#
* x * # + x(p);
x " & > # # P1, P2 " j 2S P3
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>@ % WN
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e N
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k
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k =0, 1, …N–1.
x # & > " ' =;, "@ ,
#
' ' $ .
; * # + x(p) @ -$ * (# * ) . 8.10.
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C , @ > % # =;
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# *
* &%> " $ =; $ # * # + :
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X(k )
¦
n 0
kn
x (n ) WN
N / 2 1
¦
n 0
k = 0, 1,…N–1.
138
k ( n N / 2)
x (n N / 2) WN
(8.18)
kN / 2
& , WN
N / 2 1
X(k )
¦
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e
j 2NS k N2
e jSk
>x(n) (1)k x(n N / 2)@ WNkn
= #
k (8.19) " 2k
# &$ &$ =;:
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X(2k 1)
(1) k 2k+1, & %-
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(8.19)
¦ >x 0 (n )@ WNkn/ 2 ,
N / 2 1
¦ >x1 (n )@WNkn/ 2 .
n 0
"+ =; $ # * # + & % "
=; &$ N/2- &$ # + * x0(n), x1(n), #&$ #>@ " :
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x1 (n )
>x (n ) x (n N / 2)@ WNn ,
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n = 0, 1, …(N/2)–1.
& % (8.20) > "* $ # , # * " + %& +&
=; % ( . 8.7). D '
« » "> , % & ' % – & .
" # + * x0(n) x1(n) % %
+ # (N/4)- & # + , =;
&$ %
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"> =; $ # * # + X(k).
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& # +&$ . D & > %, $ # * # + x(n) # & =; % . , $%% "#+ '* $ . J " *
# + * +> # .
139
= &* +&* =; % " +& % + =; % ( . 8.9 # N = 8).
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+" +
-$ * . 8.10.
. 8.12. < -
! ! ?#D
140
D =; > # > & +> E +, #> (8.17).
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+
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x(n)N
=;N[x(n)]
% h(n)N
=;N[h(n)]
% X(jZk) Y(jZk)
=;N[x(n)]
% y(n)N
H(jZk)
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A @+> =; % , >@ $ #, & > =; $ # * # + x(n)
+ * $ h(n), @+> =; % & D=; $ " # Y(k), >@ $ # &* # # #
- &* # . &$ #& & D=; E > #,
"+ +> > $ # +
&$ & ' .
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% +, & =; % +" + #
E & D=;, # # + x(n)
# x (n )
1
N
N 1
¦ X(k ) WN kn , n = 0, 1,…N–1.
k 0
= # %#& E & % > ' > % ( *), :
*
^
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ª N 1 *
kn º
1
1 =; [ X* (k )] * .
x (n )
« ¦ X (k ) WN
(8.21)
»
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N
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! " , & & + D=;, % * X(k) X*(k)=Xre(k)–jXim(k), " " # * >@*, & + =; # + X*(k) & " + 141
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X(k) &$ # * x(n) # + *, % 1/2
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$ $ =; # , "#>@ * & " ; + – =; D=;.
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+ # +> E > & * .
= E " %& # [21].
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x1(n)N/2 x2(n)N/2, # $
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& =; "#> :
X1 (k )
>
@
1 X (k ) X* ( N k ) , X ( k )
2
2
>
@
1 X ( k ) X* ( N k ) ,
2 j
X( N) X(0) .
k = 0, 1,… (N/2)–1.
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1.
2.
3.
4.
5.
6.
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& # " ; + (=;) % .
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= =; + ". D # # =;.
= =; + ' .
& D=; @+> =;.
142
9. ! "" =
# "# +" & #>@ : [8, 14, 15].
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% &# + &, " &* %&. ! , " % & @ > @+> "', $*: % , & , % .
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, # & +& ' $ # # + $.
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143
D$$ # " +" " "&$ ' +&$ : , , % *, < , #<
, , * "# % , * ,
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x(n)
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X(n-1) X(n-2) X(n-3)
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& # # + " &$ .
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& (A++, Java,
Pascal), # ' '
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" .
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144
C" &$ &$ # " " + & +& " % E * "&
+ "' > DA ' .
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30
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+ ( /)
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4800
8000
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MFLOPS (Million
Float Operations Per Second) # '
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# %
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TMS320C2xxx
TMS320C5xxx
TMS320C6xxx
ADSP-21xx
! (K')
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=
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20–40
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Motorola DSP56303
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TMS320C6201
=
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100
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1000
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60
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10
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& %% ;
Analog
Devices
Motorola
Texas
Instruments
= ' &
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ADSP-21xxx
DSP5600x
DSP563xx
DSP96002
TMS320C2xx
TMS320C3xx
TMS320C4xx
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TMS320C54x
TMS320C5000
TMS320C662x
TMS320C67x
TMS320C8xx
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16
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32
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24
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147
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40
40
80
20
40
25
30
50
50
40–2000
1200–2400
600–1000
50
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x "' * ' % (
+&$ " # *);
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148
x +< + + #&$ + ;
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& + * &. J > % " % " % * ,
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>
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i 0
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#&, +"&
= # &< " # + &. J # >
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* .
?* (1903–1957) # % '' > & + * < & (
, $ * &), % +< &$ < .
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+ # & ' MAC).
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@ " & * , . . $ & + # "
== < % + " #&$ = 151
< ¥=. A , E # & '
MAC # ' & ' . C+ " " &$ # +&$ # ' MAC # # ' . C" & & "'
'
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' & &> +< * * +> ==, =.
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$ #&$ ( , ' E
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F# .?. D' #
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Eyre J., Bier J. The Evolution of DSP Processor / IEEE Signal Processing magazine, 2000, March.
°. ? =A Texas Instruments / & $ , 2001. – ¢ 1.
167
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20. * .A., K.=. C# $ '
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¦ h[n] e jZnT# ,
H( j Z)
(7)
n 0
# * * "# * * $ Hd(jq) * $ * (; +z " ) * ' W(jq):
H ( j Z)
W ( j Z) * H d ( j Z)
T#
2S
Z# / 2
³ W( j T) H d [ j (Z T)] dT ,
Z# / 2
# * – , £ – W ( j Z)
,
N 1
¦ w[n] e jZnT
#
– $ * '
.
n 0
& " * * >
> . 3, # # %>@ ' "# * *
$ & # ; +.
|Hd(jZ)|
1
hd(n)
O/S
S/O
Z
n
Zc
-Zc
|W(jZ)|
wR(n)
1
G.max
Z
n
'Z
|H(jZ)|
N–1
1+G1max
h(n)
G2max
Z
n
N–1
2
'Z
N–1
. 3. < + G,D " ( % DGC "! % " )
H $ * '
<
* ¤q & , 275
. 3 &*
+ &$ $ -
" +& #> " ~.max @#+> # & . A * @ @ #$ ± q#/2 " + % * * $ * '
& @#
& "# * * $ * Hd(jq).
" #, $ # * $ + H(jq) # <
* * $ * ' : 'Z | 'Z , < ' (+' ) "# % ~1, ~2
"& &$ . J # * ' , # % +:
5. +> <
¤q;
6. +&* + &$ ~.max
+>
@#+ # & ;
7. +> # N.
! E # &. !, # & '
> +< * + &$ , +<> <
, +<>@> # &
* ' N. J G " +"&$ &$ ' *.
A# +, # &$ ' * > * + ;HF ""#& + # * * * E # + * $ :
h(n)=h(Nz1zn).
2.3.2. '" * + $ . 1 #& +"& " ; & &$ ' *: + *, + *, F, FE E.
" * <
& ¤q=Dq#/N, # D z
"& &* D- , + &$ ~.max >> % ' & " < ' * $ "# % ( +& +' * $ ) |~2max|, #, &
# '
;?H * " O c Zc T# S / 4 . ! % < > " ;H.
; # " (==;, =U;, K=;) " &$ #&$ < + ' %
&+ +< ' " , 6 #.
276
! ' 1
# " ! = +
! +
F
FE E
¤q=Dq#/N
2q#/N
4q#/N
4q#/N
4q#/N
6q#/N
~.max, #
z13,6
z27
z31
z41
z57
~2max, #
z21
z26
z44
z53
z74
` 1. C # + #& . 1, "$ >
* $ "# % ", % #+
& * ' .
` 2. & * * '
"# * $ # *
& * $ + 'f f " f min -
%& < ¤f=¤f =Df#/N $ # $ # # * '
# > # + *
$ + :
N t D f # / 'f ,
# D – E
' , " @ * * ' (D- ),
. .1,2.
U N %*< ' , &
.
` 3. " "# * * $ +"> $ & " f , @& "# % $ # * & + ¤f J " * & # # "& ' $ # & + "# % ( . 3). ? , # =;: f 1 | f 1 'f / 2 ; f 2 | f 2 'f / 2 .
` 4. ?$ # + $ + @ * (Nz1)/2 + * $ hd(m):
h (m) h d [m ( N 1) / 2] w (m), m 0,1,..., N - 1 .
` 5. C & HF
H( j Z)
+ N 1
¦ h[k] e jZk T
#
k 0
$ #& #& * $ A "$ > "# % A".
277
` 6. ! #&* # $ #&$
&$ #&$ ( '), $ # > " &$ " f1 , f2
# & + N
& >.
2.3.3. ' ', +*
&*4 + – $+% z +> <
+&* + &$
.
wR(n) = 1, n = 0,..N – 1.
(8)
;+% + * # $ +&$ &$ ' * # * N/2:
w T (n )
­° 2 n ,
w R (n ) * w R (n ) ® N 21 n
°̄2 N 1 ,
0 d n d N 1
N 1 n d
2
2
N 1
(9)
# +< <
# +< &$ .
& > <
¤q=2q#/N ¤=4/N.
!""@ + HK & & % w H (n ) D (1 D) cos( 2 S n ) .
(10)
N 1
= =0.5 * + H, * + HK.
+ &$ * ' FE & # $ % * ?C;.
& * $ >
¤q=q#/N ¤=2/N. = @#+ # & 0.04 % @# # * $ ' .
+ K #
w B (n ) 0.42 0.5 cos( 2 S n ) 0.08 cos( 4 S n ) .
N 1
N 1
=0.54 –
"& <
*
(11)
= > * ' * FE < *
&* ( 1.5 ") + &$ .
¥
&$ E * * ' ¤q=q#/N ¤=2/N.
=
" ?C; +"> % E & '
+ C4, %-^"'4, $$,
# . [7, 8],
# &$ " &$ ' * *" .
278
' + *.
# $ &$ ' *, $ ">@ $ & " &$ ~.max
< D
'f f#
N
'f f#
N (D- ), &$
' * *" E -
& <
+
+ @+> K _, $ #@ & % E * ' :
> @
2
w A (n ) I 0 (E 1 2 n / I 0 (E) ,
N 1
(12)
# I0(x) z ' #.
# E < # # #
" ' "# * * $ +< * # + "# ' .
*" ' (. 2) & E
&, & " > # "# "$ > "=|~2max| (#) *
$ H(jq), >@* #+&* ;?H, & +
+ " D- E
' & ¦ [5]:
D|
A " 7.95
,
14.36
A " ! 21 #; D 0.9222, ­0,
°
0.4
E ®0.5842 (A " 21) 0.07886 (A " 21), °0.1102 (A 8.7),
"
¯
A " 21 #;
A " d 21 #
21 A " 50 #
A " t 50 #
= & " " '& " > D # $ # &* # + N§Df#/¤f , &* " # %*< +< .
# # $ &$ ' *, '
#+&$
+
==;, =U;, K=; "$ * $ "# % % &+ +< " , 6 #.
! ' 2
A", #
25
30
35
40
45
50
55
60
¦
1.333
2.117
2.783
3.395
3.975
4.551
5.102
5.653
D
1.187
1.536
1.884
2.232
2.580
2.928
3.261
3.625
", #
65
70
75
80
85
90
95
100
279
¦
6.204
6.755
7.306
7.857
8.408
8.959
9.501
10.061
D
3.973
4.321
4.669
5.017
5.366
5.714
6.062
6.410
. 3 #& % & " +' *
* $ , >@ " & " "$ "# % [5].
! ' 3
A", #
30
40
50
60
1 ±~1max, #
±0.27
±0.086
±0.027
±0.0086
A", #
70
80
90
100
1 ±~1max, #
±0.0027
±0.00086
±0.00027
±0.000086
2.3.4. $+%' , %', ]
+&$ $ ; " >
@ & " ; + $ # "
&$ &$ $ HF Hd(jq).
% E^, " &<, +
$ # & % Z c T#
S
h d (0)
Oc
S
; h d (n )
O c sin(O c n )
,
S
O c n
n=r1, r2, …,
(13)
% $$+?@ % (=;) &$ # # $ #:
y(n)=x(n); hd(0)=1; hd(n)=0 nz0; H d ( j Z) 1 Z d Z# / 2 .
(14)
+& $ ; ^, & ( ),
U ( % ) V& ( ) &+ & %&
" +& $ '
E^
&:
H d ( j Z) ;H H d ( j Z) =; H d ( j Z) ;?H ,
(15)
H d ( j Z) =;
H d ( j Z) C;
H d ( j Z) ;?H 2 H d ( j Z) ;?H1 ,
(16)
H d ( j Z) =; H d ( j Z) ;?H 2 H d ( j Z) ;?H1 ,
(17)
Hd(jq);?H2 – & $ # Hd(jq);?H, Hd(jq);?H1
#+&$ ;?H " c, c1, c2, (c2> c1), >@ " ;H, =; C;.
! % "+ # # +&$ $ , " # " + >@ & % :
h d (0) ;H
1
Oc
S
, h d (n ) ;H 280
O c sin( O c n )
,
S
O c n
n=r1, r2, …
(18)
h d (0) =;
h d (0) C;
O c 2 O c1
, h d (n ) =;
S
S
1
O c2
O
c1
S
S
, h d (n ) C;
sin(O c 2 n ) sin(O c1 n )
S n
S n
,
sin(O c1 n ) sin(O c 2 n )
.
S n
S n
(19)
(20)
& " $ # < # K=;.
2.4. U H-% * '"
2.4.1. !" * '"
# * & + $ + h(n)N $ # # "'
"# * *
& # $ Hd(jq)
" ; + (D=;).
"' * $ Hd(jq) @ 0 … q# $ # & &$ " * &
q # &: qk=¤qk, # k=0, 1, …, Nz1; ¤q=q#/N z < # "' ; k z * & ; N z # "' .
` ¤q & " ¤qu¤q /(L+1), # Lz'& , L = 0, 1, 2, …; ¤q z $ #
+ .
"+ # "
$ + (HF) H d ( j Zk ) H d ( j Z) Z Z ( . 4). ! "#
k
$ " " + & ""#& , # ; "&
HF # "
$ %# # $ # "
* HF.
"' * $ . 4 & < ¤q=¤q /2 (L=1).
. 4. E CA ! % HF " , & 1 (Hd(jqk)=1),
$ # * – "# % z > (Hd(jqk)=0)
281
& % & + & ( " &) " Hd(jqk)=H1=var, &$ " ' "# *
* $ .
HF Hd(jqk) % + > +> $ hp(n), #> @+> # " ; + (D=;):
h p (n )
1
N
N 1
¦ H d ( jZk ) e jZ k n T# .
k 0
= + $ ( . 5.) * # * # Np=N, . . # "'
# "' * .
+ * $ " # * & ?C; & # # + * $ hp(n), # &* (Nz1)/2 (# " * " ) &* + * *
' * (# F- + ) ( . 5.):
h (n )
h p (n N 1), n
2
0,1,...N - 1 .
)
)
. 5. @" % , "+ ECA ( )
" % G,D,
! ()
= + * $ h(n) $ # $ + H(jq), >@ "#> Hd(jq):
H( j Z)
N 1
¦ h(n) e jZn T#
n 0
HF + $ q=qk: H(qk)=Hd(qk) # & & HF, $ qvqk H(q)vHd(q) z "# * < ' . ;HF + * # + * $ .
$$ '" * , $,* $ L
$ " *
Hi. (i=1,2,…,L), #>@ $ > ' > # *.
282
C" & " L > #>@ + &$ :
L = 0: ~2$ § z20 #;
& "-
L = 1: ~2$ § z40 #;
L = 2: ~2$ § z (50 z 60) #;
L = 3: ~2$ § z (80 z 100) #.
C+ # * & % "
+ ?C; +& "$ "# % # (90z120) #.
! " , "' + ">
& L z
& $ # * $ +&$ " *
Hi. , " >@ $ < ' . D # , + &$ & @ %
'# "' . D # E " JK # * .
2.4.2. & * '"
` 1. = " > "# "$ "# % " & + &$ L * $ $ # * . ? , " u 40 #, L = 1.
H % HF + , +< "$ # " L.
` 2. " L "# * $ # * &
'f f " f $ # < # "'
* $ : 'f
'f L 1
# "'
: N
f#
'f
L 1
f#
'f .
=
N %*< ' , & .
` 3. " "#> > $ Hd(jq)
< ¤f, "+ HF Hd(jqk), k = 0, 1, …, Nz1.
D # k # &$, &$
+ &$ &$ & .
U# +& " Hi. " &$ &$
& %# * $ # * , , * * ' HF %# & " "# % .
` 4. C & > $ ?(jq) $ # " Hi. , &$ $ # "#& .
? , # ;?H
L = 1, N = 33 " H1 =0.3904, ~2max= z40 #;
283
L = 2, N = 65 H1 = 0.588, H2 = 0.1065, ~2max < z60 #.
` 5. C & +> $ ?C; * $ :
h (n )
H d ( 0)
N
( N 1) / 2
¦ 2 H d ( jZk ) cos> n N21 Zk T# @
1
N
k 0
n = 0, 1, 2, …, Nz1.
2.5. ^' ' ', %
H & +& #& " ; "> JK @+> '# # * ' "#&$
&$ $ + #&
"'
< ' . = E &
$ + + " +>
. D &
'
F F- +
> + 4" (AD) +4 "'4
$" ('* *).
* ! #>@> ' > ' >
M
E
¦ > H( j Zk ) H d ( j Zk ) @2 ,
(21)
k 1
# H d ( j Zk ) , H( j Z k ) – "# >@ & $ + , & & # % qk. J ' * + E
' + .
V'* * "> "' % +&$ " * " < ' < :
E (Z)
W (Z) H ( j Z) H d ( j Z) ,
(22)
# W(q) – % + ' .
= +&$ " * E
' + * '
@ # +< $ # ,
* , * * "'
(
; -=E # F- + ) * "& C"
(#
+
&< * ' * F F- ). $ > E & +> & &, , K " +&$ > H&< F+ , +& & " ; FDAS2K, DFDP, Signal & MatLAB # .
284
2.6. + Simulink $ ',
H–%
C MATLAB @ @+> &
+ Simulink. U Simulink %
" " # MATLAB, %
& > #+ ( ).
= " Simulink & > # : untitled
( # "# –# & # )
Library Simulink
( ) &$ "# .
& < untitled $ # # + , # >@ , " +&$ &$ .
" $ " $ # & + # * * @ . = E # % &+ * Block Parameters.
2.6.1. %
#
& '
+ (;) #>@> > $ ( #+) ( . 6).
. 6. " " ! %
K #+ '
+ "# @+> Digital Filter
Design ( . 7), (DSP Blockset/Filtering/Filter Design/ Digital Filter Design).
. 7. 285
Digital Filter Design
$ #& #& ; "#> Parameteters: Digital Filter Design ( . 8).
+ Block
. 8. ? Digital Filter Design
'
+ E& * & 6 :
x Current Filter Information – % ' " '
+ ( # – Order, * + –
Stable/Unstable, – Sections, &
+ – Filter structure);
x Filter Type – "# + :
- Lowpass – ;?H;
- Highpass – ;H;
- Bandpass – * + =;;
- Bandstop – % &* + C;;
- Differentiator – #
' &;
- % # & + ;
x Design Method – "# # ' :
- IIR – F- + &:
- Butterworth – + ;
286
x
x
x
- Chebyshev Type I – + H&< 1 #;
- Chebyshev Type II – + H&< 2 #;
- Elliptic – + E * (U -E );
FIR – F- + &, . $. 2.3–2.5:
- Equiripple – &* * ( &*),
$. 2.5;
- Least-squares – * + AD, $. 2.5;
- Window – # &$ ( &$) ' *, $. 2.3;
- Filter Order – "# # + - (Specify
& % + order) # + - (Minimum order);
Frequency Specifications – "#> & & + ( % "+ " & + ):
- Units – # '& " & (Hz – ', Normalized (0 to
1) – " &* + ( +&$ # '$);
- Fs – # "' ;
- Fstop1 – % & " %# ( *
"$ Astop1, #);
- Fpass1 – % & ( *
"$ Apass, #);
- Fpass2 – $ & ( *
"$ Apass, #);
- Fstop2 – $ & " %# ( *
"$ Astop2, #);
Magnitude Specifications – "#> E
' & "$ + :
- Units – # '& " E
' "$ (dB – #,
Squared – +& # '&);
- Apass, Epass – E
' & "$ ;
- Astop, Estop – E
' & "$ " %# .
=
& Equiripple # ( &* *) $ # # + "#+
Options
( . 9) Density factor, > &* 16.
=
& Least-Squares + AD $ # # + "#+ Magnitude Specifications ( . 10)
& E
' & $ "# % Wstop1,
Wstop2, Wpass, > & 1.
287
. 9. * Options
. 10. * J
!" Magnitude Specifications
=
& Window # &$ ' * $ # "#+
Options ( . 11) * ' Window, % # &$ ' * # +& &, Beta
# * ' *" Kaiser.
. 11. $ " Window
D & & ' :
x Bartlett – ' ;
x Blackman – ' E;
288
x
x
x
x
x
Hamming – ' FE ;
Hann – ' F;
Kaiser – ' *" ;
Rectangular – + ' ;
Triangular – + ' ;
% "+
,
& # " &$ #* * Digital Filter Design + , # %@> & #>@ " :
"#+ &* * &+ * $ + * + # % ;;
;;
;;
;
# # + # +> # % + # #*
+ < "#;
;
;
# % ;
+< # % ;
& #
#+ Filter Visualization Tool "
+ .
D Filter Visualization Tool " "+ +
& + , :
HF
+ ;
;HF
+ ;
#
# + $ ' );
HF
+ ;
;
+ *
;HF
>
"#*
( $ # $ -
+ * Z- ;
" E
' # * '
+ (Numerator –
' & ").
E
' & , Denominator – E
289
2.6.2. Gain (+%)
= + Digital Filter Design " & + + & + &, . . # & , # E
' # , >@ $ # '&, $ # +" + # +&* Gain (Simulink/ Math/ Gain) ( . 12).
. 12. E
' "#
Block Parameters: Gain ( . 13).
Gain
*
. 13. 290
Gain
2.6.3. Signal Generator (+%'* )
, & #+ $ #
+ , +">
+&* Signal Generator (Simulink/
Sources/Signal Generator) ( . 14).
. 14. Signal Generator
. 15. 291
Signal Generator
* ( . 15) Signal Generator "#> #>@ &:
x Wave form –
:
- sine– #+&* ;
- square – +&* ;
- saw tooth – "&* ;
- random – *&* (<);
x Amplitude Frequency – # ;
x Units – # ' " & (Hertz – '& rad/sec –
#/).
2.6.4. Zero-Order Hold (+* '"-,,
H)
H & $ # ; #+ '
* , +" F, " &* &$ # (Signal Generator) # & & . F +"
Zero-Order Hold (Simulink/ Discrete/ Zero-Order Hold) ( . 16).
. 16. Zero-Order Hold
* Zero-Order Hold "# "' Sample time ( . 17).
292
# # -
. 17. Zero-Order Hold
, " . 17, # # "'
# * # "' , . . 4000 '. A# +, # "' , & Zero-Order
Hold, # % &+ # "' Fs, " * '
+ ($. 2.6.1).
2.5.5. Step
Step (Simulink/ Sources/ Step) ( . 18.), +" #
" # "#* .
)
. 18. )
Step ( ) ! ()
=
&
* ( . 18.),
Step time "# "#* , $ Initial value Final value – + " #& "#* , Sample time – # # 293
"'
&$ # ( # >, & &).
H & * + # "#* ($,+? ,+), % + #>@>
> $ ( . 19).
. 19. " " %
2.6.6. Scope ()
"+ +"> , &
#
> + &$ ; % Scope( ' ) (Simulink /Sinks/ Scope) ( . 20).
. 20. Scope
>#+ ' %&$ Scope
# $ # " ' #
>@ +" '&. D & $ # (. . , # % %+ # 30 294
). " ' + + E . J ' # $ #
%
&-
# , " ).
. 21 ( &# . 21. ' ! Scope " ?% &
# > *
Scope (Scope parameters) ( . 22).
. 22. 295
Scope
Number of axes "# $ #
' ,
Time range – $ * # , % ' ,
Tick labels – % * # (all – , none – *,
bottom axis only – + " + +). C # +" + * , & & >.
2.7. U' $ Simulink (?
Simulation)
? , > Simulation ( #
) # % + # ( . 23), > > +
#
# * # . = # E $ # " " % + + # + #
, "+ %*< & # , , , " #+ ,
# "+ #
.
. 23. I+ Simulation
C # # #+> #
, & # # > Simulation Parameters ( & #
) ( . 24). J&
# Solver & &.
Simulation time ( #
) – &
#
# " + (Start time) (Stop time) " * #+ .
Solver options ( & ) – &
# "'
( ) # .
296
. 24. " Output options ( & & #) – & & # &$ #&$ # * & ( #
& < ).
= # & # "' # # #>@. # * &
# –# &,
" % & + # % $ # #
.
A @+> # $ #& >@ $ Type (! ) %
&+ " #>@ $
$:
# & $ # "
x # & # # ;
x # & & & $ #;
# & $ # ;
x & & & & $ # .
x & & = &* ( ) " & + " #+ :
x Variable – step ( &* <) – #
&
< ;
x Fixed – step ( &* <) – #
& < .
* ( ) " & + # &. = &* (discrete) # &$ * &. D+& & >
& # # & &$ . J
#& " > # (Variable – step) #
297
(Fixed – step) < , & # * # –
< & &$ #
' +&$ *(ode).
? % # $ & >@ $ Type $ # , " " " & " #+ ( # + & >).
= & # $ # % % +" + >.
3.
=CDCKK?D DA=H?
=
& * & +"> &
MATLAB 6.0
&<, % Mathcad 2000 &<.
4.
=CDCKK \DC!DC?D³ CD!µ
4.1. " + " F- +
# &$
' *, * & , % & # .
4.2. & + Mathcad " F- + # , + E
' & # *
' ;, + & " "#&
. ?* $ #> $ + .
4.3. & +
MATLAB " F+
# &$ ' * *" , >
+ AD
>. C +
E
' & # * '
;, + HF "#& . ?* $ #> $ + . ?* + & "# % ,
' + " +& * + .
5.
D?!CD\²?µ D=CDAµ
5.1. H + $ # ' ;?
5.2. H # ; "& " "'
+ ?
5.3. D $ " $ ;
# ? " + &
+ ?
5.4. # F- + # &$ ' *?
5.5. & % & > +& $ #+&$ ;: ;?H, =;?
5.6. " " & * #
# < (AD) ; & # ?
298
5.7. & #&
" # #
+"> MATLAB # F- + ?
5.8. & + " "+ + Filter Visualization Tool Digital Filter Design?
5.9. +" Gain #
&
;?
5.10. " Zero-Order Hold #
& ;?
5.11. " "# # # "'
ZeroOrder Hold # % &+ ?
5.12. & $ # +, & &$ #
Step + & /# "#* ?
6.
=DCD µ=D\?? \DC!DC?DD U?
6.1. = # +
$ #& #& ($. 2.2.1)
"#& ( +> ),
$ # & # " E^: – + $ N- ($. 2.2); " F; E
' K0; # "' F#=16 '.
F
K0
1
100 '
10
2
200 '
20
3
300 '
30
4
400 '
40
5
500 '
50
6
600 '
60
= # + # + N # * &
" E
' ($. 2.2.2–2.2.5).
D' + & " + "# *.
= "+ E
' & # * '
HF.
6.2. A "#+ #+ + ($. 2.6.1) Simulink.
= # +
$ #& #& ($. 2.6.1) "#& ( +> ), $ # &
# " $ % (&): # – ' + * ($. 2.3.3, 2.6.1); ' + &
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F
K0
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4
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6.5. = #+ $ # ; * ($. 2.6.3) *
Fx * Fc
# * # *. = +
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6.7. &% $+' 6.2 – 6.6 # '
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E
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F
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2
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3
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4
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5
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6
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H # "' Fs + * 16 '.
= # + # % &+ #
+ , # &$ ' * *" , $. 6.2–6.6.
6.8. &% $+' 6.2 – 6.6 # '
+ (=;), > + AD Least-squares ($. 2.6.1, 2.5): ' + & F; E
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Fstop1=0.5 F 60 #; Fstop2=2 F – 80 #.
F
K0
1
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10
2
200 '
20
3
300 '
30
4
400 '
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5
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6
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= # + # % &+ #
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A=AD \!C!Cµ
1. .A. : . 2 . H.1. – : "#- !, 2001. – 199 .
2. + .. "+ #
# MatLab.
. . – A=.: = , 2000. – 480 .
3. + .. MatLab. ' #
# Windows: . – A=.: DCD? = , 1999. – 288 .
4. +# \.K. # . . – 2-
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5. C \., # . !
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* . / = . . # #. °.?. # . – K.:
K , 1978. – 848 .
6. A .. . – A=.: = ,
2003. – 604 .
7. ., # ., J =. &
+ &
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8. FE C.. & + &. – K.: ?# , 1987.
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1.1.
1.2.
1.3.
1.4.
1.5.
D & &
# .......................................................... 4
& & .................................................................. 9
= $ #& # & ................................................ 14
& *& & ............................................................. 16
+& & ............................................................................ 19
2. ;CDµ A?\µ AA!Kµ ....................................................... 20
2.1. C & '
* ............................ 20
2.2. K # # &$ ................................... 22
2.3. A # .................................................................. 23
2.3.1. A "+ %# # , % ............. 24
2.4. D # ' # &$ ............................ 29
2.5. K #& *&$ # &$ * & '
* + '
$ ............................................................................................. 30
2.6. K #& # &$ * ( * ) ............................... 33
2.7. ! & # + # &$ ............................. 37
2.8. = # ' $ # * & ................................................................................ 41
2.9. = # & '
&$ '
&$ + .
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2.10. H & $ &$ + .
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............................................................................ 49
2.11. ; & "'
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2.12. = "' &$ +
......................... 58
2.13. = # ' $ + ......................................................................... 60
303
2.14. ? & + & * * " *
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2.15. = & < "# '
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2.16. +& & ............................................................................ 67
3. =DC¥?DA! ?!D? A?\D ....................................... 68
3.1. A $ < .............. 68
3.2. = < + " &$ " $ # ............................................ 70
3.3. = < + " # # # " * & ........................ 74
3.4. D # < ## * < +> " ............ 76
3.5. +& & ............................................................................ 77
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4.1. # # "
,
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4.2. & " # "
..... 80
4.3. ! .. + ................................................................... 83
4.4. & ,
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.................................... 88
4.6. +& & ............................................................................ 89
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5.1. U#
#& " '
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5.2. A " &$ +
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5.3. K # * " .................................................... 93
5.3.1. D@ # ............................................................... 93
5.3.2. * " ........................................................ 94
5.3.3. K # " C; .................. 96
5.3.4. C " ;?H- (;=?H) ........................ 98
5.3.5. = $ # ;=?H ; "# ............................... 100
5.3.6. = " + # * " .................................................... 101
5.4. +& & .......................................................................... 104
6. ;CDµ F-;\²!Cµ ..................................................................... 105
6.1. A " &$ +
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6.1.1. D # ......................................................................... 105
304
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6.1.3. & ' *" ......................................................... 110
6.1.4. +& $ #+&$ ; " ................................................ 112
6.1.5. K # " ?C; # &$ ' * ............. 113
6.2. A " &$ +
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6.2.1. D # ......................................................................... 114
6.2.2. K # " ?C; # * & .......... 118
6.3. +& & .......................................................................... 119
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7.1. " '
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8.1.
8.2.
8.3.
8.4.
8.5.
8.6.
8.7.
8.8.
8.9.
D # * =; ............................................................. 122
; + ' =; ................................................ 126
A +&* " : "# , #&, & .......... 128
A +&* " =; ................................ 129
D # ' =; .............................. 131
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=; > 2 % ......... 138
& D=; =; ............................... 141
+& & .......................................................................... 142
9. A!CD³A! ;CDD³ DCD! A?\D ........................ 143
9.1.
9.2.
9.3.
9.4.
A & "' DA ............................................... 143
D DA, >@ E> " ..... 145
D & * =DA ................................................................... 148
D@ ' & $ =DA ........................ 150
9.4.1. = $ ' ......................................... 150
9.4.2. $ ?* # $ .......... 151
9.4.3. A ' =DA .................................................... 152
9.4.4. D & '
=DA ............................ 155
9.5. D & & =DA ......................................................................... 159
9.5.1. ' '
=DA ......................................... 159
9.5.2. A# & ' & =DA (Conventional DSP) ............. 159
9.5.3. <& # & ' & =DA
(Enhanced-conventional DSP) ...................................................... 162
9.5.4. = ' & =DA $ * VLIW ............................... 163
9.5.5. A & ' & ..................................................... 163
9.5.6. #& ' & .............................................................. 164
305
9.6. $ & " % ' .......................... 165
9.7. +& & .......................................................................... 165
U\°H? .................................................................................................. 166
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\ ¢ 1
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\ ¢ 2
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\ ¢ 3
K #
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\ ¢ 4
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\ ¢ 5
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