数字信号处理：第四章k

4-5
4-3
4-4
4-2
4-1
1,,1,0,)()(
1
0

NkWnxkX
N
n
nk
N?

1
0
1,,1,0,)(
1
)(
N
k
nk
N NnWkXNnx?
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1210 )1()2()1()0()1( NNNNN WNxWxWxWxX?
.
nkNW;
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kNn
N
nk
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nk
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nk
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WW


.
),1(1
),1(
)2/(
2/
)(2)()(
2
k
N
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N
j
N
N
N
nkNn
N
Nk
N
nk
N
knN
N
kNn
N
WW
eWW
eWWWWW
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—
1,,1,0 ),()12(
1,,1,0 ),()2(
22
21


N
N
rrxrx
rrxrx

1
0
)()]([)(
N
n
nk
NWnxnxD F TkX
)(nx






1
0
2
2
1
0
2
1
1
0
)12(
1
0
2
1
0
1
0
22
22
))(())((
)12()2(
)()()(
NN
NN
r
rk
N
k
N
r
rk
N
r
kr
N
r
rk
N
N
n
N
n
nk
N
nk
N
WrxWWrx
WrxWrx
WnxWnxkX
n
2
22 )/(222 NNN WeeW jjN
)()()()()( 21
1
0
2
1
0
1
2
2
2
2
kXWkXWrxWWrxkX kN
r
rkk
N
r
rk
N
N
N
N
1 2
1 2
k
N




1
0
1
0
22
1
0
1
0
11
2
2
2
2
2
2
2
2
)12()()(
)2()()(
N
N
N
N
N
N
N
N
r
rk
r
rk
r
rk
r
rk
WrxWrxkX
WrxWrxkX
1,,1,0 2?N?
rkkr
N
N
N WW 2
2
2
)(
)()()()
2
( 1
1
0
1
)(
1
0
11
2
2
2
2
2
kXWrxWrxkNX
N
N
N
N
N
r
rkkr
r

)()2( 22 kXkNX
kNkNNkN WWWW NN 22 )(
)2()2()2( 21 2 NkXWNkXNkX NkN
1,,1,0 ),()( 221 NkN kkXWkX?
N/2 )
1 1
1
1
)()()(
)()()(
21
21
kXWkXkX
kXWkXkX
k
N
k
N


)1,,(
)1,,1,0(
2
2


Nk
k
N
N
)()()( 21 kXWkXkX kN
)()()2( 21 kXWkXkNX kN
)(1 kX
)(2 kX
kNW
X1(k) X 2(k) X(k) -
4)2(
22 NN?
)12(2?NN
2
2N
)12(?NN
2
N
NN 22
2)12(
2NNNN
222
22 NNN
*
)(42/ 1 kXD F TN?
3,2,1,0
)2()()(
3
0
4
3
0
411


k
WrxWrxkX
r
rk
r
rk
);6()3(
),4()2(
),2()1(
),0()0(
1
1
1
1
xx
xx
xx
xx
);6(),4(),2(),0( xxxx
);7()3(
),5()2(
),3()1(
),1()0(
2
2
2
2
xx
xx
xx
xx
3,2,1,0
)12()()(
3
0
4
3
0
422


k
WrxWrxkX
r
rk
r
rk
)(42/ 2 kXD F TN?
3,2,1,0),()()4(
)()()(
21
21


kkXWkXkX
kXWkXkX
k
N
k
N
x x
x1 x
x1 x
x1 x
x2 x
x2 x
x2 x
x2 x
1 2
~ ~
1
1
1
1
2
2
2
2
WN2
WN1
WN0
WN3
-1
-1
-1
-1
L
1,,1,0),()2( 431 Nlxlx?
1,,1,0),()12( 441 Nlxlx?
kl
N
l
lk
N
l
lk
N
l
lk
N
l
WlxWlxkX
WlxWlxkX
NN
NN
)12(
2/
1
0
14/
1
0
44
2
2/
1
0
14/
1
0
33
)12()()(
)2()()(
44
44


)()()( 431
2
kXWkXkX kN
1,,1,0 4 Nk?
N/4 X1(k)
)()()4( 431
2
kXWkXkNX kN
N/4 X1(k)
1,,1,0 4 Nk?


1
0
4/64/
1
0
26
1
0
4/5
1
0
4/25
44
44
)()12()(
)()2()(
NN
NN
l
lk
N
lk
N
l
l
lk
N
l
lk
N
WlxWlxkX
WlxWlxkX
14,,1,0k; ( k )XW( k ) X( k ) X 6kN / 252 N?
14,,1,0k; ( k )XW( k ) Xk)4N( X 6kN / 252 N?
)()( 65 kXkX
)4()2()1(
)0()0()0(
)()()(
13
13
13
xxx
xxx
nxrxlx



(1) x(n),
N/4 DFT X3(0),X3(1)
)6()3()1(
)2()1()0(
)()()(
14
14
14
xxx
xxx
nxrxlx



(2) x(n),
N/4 DFT X4(0),X4(1)
(3) x(n),
)5()2()1(
)1()0()0(
)()()(
25
25
25
xxx
xxx
nxrxlx



N/4 DFT X5(0),X5(1)
(4) x(n),
)7()3()1(
)3()1()0(
)()()(
26
26
26
xxx
xxx
nxrxlx



N/4 DFT X6(0),X6(1)
5 X3(0),X3(1),X4(0),X4(1),
X1(0),X1(1),X1(2),X1(3)
6 X5(0),X5(1),X6(0),X6(1),
X2(0),X2(1),X2(2),X2(3)
3 1
3 1
4 1
4 1
5 2
5 2
6 2
6 2
0
2
N
N
0
2
N
N
W
W
W
W
0
1
2
3
N
N
N
N
-1
-1
-1
-2
-1
-1
W
W
W
W
3
3
4
4
5
5
6
6
1
1
1
2
2
2
2
1
7 X1(0),X1(1),X1(2),X1(3) X2(0),X2(1),X2(2),
X2(3),
X(0),X(1),X(2),X(3) X(4),X(5),X(6),X(7)
0,1k,)()(
0,1k,)()(
0,1k,)()(
0,1k,)()(
4/
1
0
66
1
0
4/55
1
0
4/44
1
0
4/33




lk
N
l
l
lk
N
l
lk
N
l
lk
N
WlxkX
WlxkX
WlxkX
WlxkX
)4()0()1()0()1(
)4()0()1()0()0(
0
3
1
233
0
3
0
233
xWxxWxX
xWxxWxX
N
N


)6()2()1()0()1(
)6()2()1()0()0(
0
4
1
244
0
4
0
244
xWxxWxX
xWxxWxX
N
N


)5()1()1()0()1(
)5()1()1()0()0(
0
5
1
255
0
5
0
255
xWxxWxX
xWxxWxX
N
N


)7()3()1()0()1(
)7()3()1()0()0(
0
6
1
266
0
6
0
266
xWxxWxX
xWxxWxX
N
N


WN0
WN0
WN0
W 0N
-1
-1
-1
-1
3
3
4
4
5
5
6
6
WN0
WN2
WN0
WN2
-1
-1
-1
-1
1
1
1
2
1
2
2
2
W
W
W
W
N
0
N
1
N
2
N
3
-1
-1
-1
-1
x
x
x
x
x
x
x
x
,
2
W
W
W
W
N
0
N
0
N
0
N0
-1
-1
-1
-1
W
W
W
W
N
0
N
2
N
0
N
2
-1
-1
-1
-1
W
W
W
WN
N
N
N
0
1
2
3
.
.
.
.
.
.
.
.
.
.
.
x
x
x
x
x
x
x
x





r
Nmmm
r
Nmmm
WjXkXjX
WjXkXkX
)()()(
)()()(
11
11
),3()6(
),2()2(
),1()4(
),0()0(
0
0
0
0
Xx
Xx
Xx
Xx
).7()7(
),6()3(
),5()5(
),4()1(
0
0
0
0
Xx
Xx
Xx
Xx
…
…
0
001
0
001
)1()0()1(
)1()0()0(
N
N
WXXX
WXXX


0
001
0
001
)3()2()3(
)3()2()2(
N
N
WXXX
WXXX








2
112
2
112
0
112
0
112
)3()1()3(
)3()1()1(
)2()0()2(
)2()0()0(
N
N
N
N
WXXX
WXXX
WXXX
WXXX






1
223
1
223
0
223
0
223
)5()1()5(
)5()1()1(
)4()0()4(
)4()0()0(
N
N
N
N
WXXX
WXXX
WXXX
WXXX
)(),( jXkX mm
)(),( 11 jXkX mm
7(),3(),5(),1(6(),2(),4(),0( xxxxxxxx
n =00
n =10
n =01
n =11
n =01
n =11
0
1
0
1
0
1
0
1
x
x
x
x
x
x
x
x
),,( 012 nnnx
(n2)
x(000) 0
x(100) 4
x(010) 2
x(110) 6
x(001) 1
x(101) 5
x(011) 3
x(111) 7
n?
^
2 1 0 0 1 2
A(1) A(2) A(3) A(4) A(5) A(6) A(7) A(8)
x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7)
x(0) x(4) x(2) x(6) x(1) x(5) x(3) x(7)
…
=(
(N+N/2)
x?
r
NW
—
1
0
)()(
N
n
nk
NWnxkX




1
0
)(
1
0
1
2/
1
0
2
2
2
2
)
2
()(
)()(
N
N
N
N
n
kn
N
n
nk
N
N
Nn
nk
N
n
nk
N
W
N
nxWnx
WnxWnx
nk
N
n
k
N WW
N
nxnx
N
N



1
0
2
2)
2
()(
,12jN eW N kkN NW )1(2
nk
N
n
k WNnxnxkX
N

1
0
2
)
2
()1()()(
1,,1,0 Nk?
nr
n
nr
N
n
N
N
N
W
N
nxnx
W
N
nxnx
2
2
2
1
0
2
1
0
)
2
()(
)
2
()(




)2( rX
r 1,,1,0 2 Nr?
nr
n
n
N
rn
N
n
N
N
N
WW
N
nxnx
W
N
nxnxrX
2
2
2
1
0
)12(
1
0
)
2
()(
)
2
()()12(


x x
1,,1,0 2 Nr?
)2()( Nnxnx
1,,1,0 2 Nn?
n
NW
Nnxnx


)
2()(
)
2
()( Nnxnx?
nNW
)2( Nnx?
)(nx
)0(x
)1(x
)5(x
)4(x
)3(x
)2(x
)7(x
)6(x
)0(X
)2(X
)6(X
)1(X
)3(X
)5(X
)7(X
)4(XDFT
N
2
DFT
N
2
x
x
x
x
x
x
x
x
.





r
Nmmm
mmm
WjXkXjX
jXkXkX
)]()([)(
)()()(
11
11
)(1 kX m?
)(1 jX m?
)()()( 11 jXkXkX mmm
rNmmm WjXkXjX )]()([)( 11
.





r
Nmmmmm
mmmm
W
N
kXkX
N
kX
N
kXkXkX
)]
2
()([)
2
(
)
2
()()(
11
11
rNW
.
rNmmm WjXkXjX )()()( 11
rNmmm WjXkXkX )()()( 11)(1 kX m?
)(1 jX m?
rNW 1?
)(kX m
)( jX m
rNW
rNW?
)(1 kX m?
)(1 jX m?
=
1
1
rNmmm WjXkXjX )]()([)( 11
)()()( 11 jXkXkX mmm)(1 kX m?
)(1 jX m?
rNW1?
)(kX m
)( jX m rNW rNW?
)(1 kX m?
)(1 jX m?
=
1 1
rNW
rNW?
1 11
1
rNW
rNW?
,FFT IFFT


nk
N
N
k
N
n
nk
N
WkX
N
kXI D F Tnx
WnxnxD F TkX


1
0
1
0
)(
1
)()(
)()()(
nkNW
nkNW?
L
LN )2
1(
2
11
.
nk
N
N
k
N
k
nk
N
WkX
N
WkX
N
nx


1
0
1
0
*
)(
1
])(
1
[)(


)(
1
)(
1
1
0
kXD F T
N
WkX
N
nk
N
N
k
)( nx
BABAWW nkNnkN ][,][?
x
)(nx
)(nx)(nh
)(nh
)(ny)(nx )(nh

1
0
)()()()()(
L
m
mnhmxnhnxny
)(ny
)(mx
)(mh
0 1 2
1 1.1
)(mx
2 3
)( mh?
)1( mh?

3
0
0)()()0(
m
mhmxy

3
0
1)1()()1(
m
mhmxy
)2( mh?
)3( mh?
)(mx
2 3

3
0
3)2()()2(
m
mhmxy

3
0
6)3()()3(
m
mhmxy
)4( mh?
)5( mh?
)(mx
2 3

3
0
5)4()()4(
m
mhmxy

3
0
3)5()()5(
m
mhmxy
)(ny
1)( MLny
,)(ny;1)(),(.1 LMNnhnx
)()(.2 nhFFTkH?
)()(.3 nxF F TkX?
)()()(.4 kHkXkY?
)()(.5 kYI F F Tny?
FFT
FFT
IFFTx
x(n)
h(n)
y(n)
X(k)
H(k)
Y(k)
.