dsp_foehu - lec 05 - frequency-domain representation of discrete time signals
TRANSCRIPT
njenx )(
k
knxkhny )()()(
k
knjekh )()(
jn
k
jk eekh )(
jnj eeH )(
njejnj eeH )()( jeH
k
jkj ekheH )()(
k
jkj ekheH )()(
)()()( jjR
j ejHeHeH
)(|)(|)(jeHjj eeHeH
)()( dnnxny )()( dnnnh
dnj
k
njd
k
njj eenkekheH )()()(
1|)(| jeH
dj neH )(
dnjj eeH )(
)cos()( 0nAnx
njjnjj eeA
eeA
nx 00
22)(
dd njnjjnjnjj eeeA
eeeA
ny 0000
22)(
)()( 00
22dd nnjjnnjj ee
Aee
A
])(cos[)( 0 dnnAny
k
jkj ekheH )()(
k
jkj ekheH )2()2( )()(
k
jkekh )(
)( jeH
,2,1,0
)()( )2(
m
eHeH mjj
k
jkj ekheH )()(,2,1,0
)()( )2(
m
eHeH mjj
|)(| jeH
k
jkj ekheH )()(,2,1,0
)()( )2(
m
eHeH mjj
|)(| jeH
k
jkj ekheH )()(,2,1,0
)()( )2(
m
eHeH mjj
|)(| jeH
|)(| jeH
|)(| jeH
|)(| jeH
Mk
k
knxM
ny0
)(1
1)(
M
k
knM
nh0
)(1
1)(
k
jkj ekheH )()(
M
k
jkeM 01
1j
Mj
e
e
M 1
1
1
1 )1(
)(
)(
1
12/2/2/
2/)1(2/)1(2/)1(
jjj
MjMjMj
eee
eee
M
)(
)(
1
12/2/
2/)1(2/)1(2/
jj
MjMjMj
ee
eee
M
)2/sin(
]2/)1(sin[
1
1 2/ Me
MMj
)2/sin(
]2/)1(sin[
1
1)( 2/ M
eM
eH Mjj
)2/sin(
]2/)1(sin[
1
1|)(|
M
MeH j
n
n
njj enxeX )()(
Analysis
deeXnx njj )(2
1)(
Synthesis
Inverse Fourier Transform(IFT)
Fourier Transform(FT)
n
n
njj enxeX )()(
deeXnx njj )(2
1)(
deemx njm
m
mj)(2
1
deemx njmjm
m
)(2
1
demx mnjm
m
)()(2
1
2)( de mnj
deeXnx njj )(2
1)(
)()(
1 )( mndjemnj
mnj
)(
)(
1 mnjemnj
)()(
)(
1 mnjmnj eemnj
)(sin2)(
1mnj
mnj
0)(
)(sin2
mn
mn
de mnj )(
n
n
njj enxeX )()(
deeXnx njj )(2
1)(
deemx njm
m
mj)(2
1
deemx njmjm
m
)(2
1
demx mnjm
m
)()(2
1
deeXnx njj )(2
1)(
n
n
njj enxeX )()(
deeXnx njj )(2
1)(
deemx njm
m
mj)(2
1
deemx njmjm
m
)(2
1
demx mnjm
m
)()(2
1
deeXnx njj )(2
1)(
n
n
njj enxeX )()(
Analysis
deeXnx njj )(2
1)(
SynthesisInverse Fourier Transform
(IFT)
Fourier Transform(FT)
)]([)( nxeX j
)]([)( 1 j- eXnx
)()( jeXnx
n
n
njj enxeX ][)(
Fourier Transform (FT)
)()()( jI
jR
j ejXeXeX
)(|)(|)(jeXjjj eeXeX
)()()( jI
jR
j ejXeXeX
)( jeX
)( jR eX
)( jI eX
|)(| jeX
)( jeX
)()( * nxnx ee
)()( * nxnx oo
)()()( nxnxnx oe
)](*)([)( 21 nxnxnxe )](*)([)( 2
1 nxnxnxo
)()()( jo
je
j eXeXeX
)](*)([)( 21 jjj
e eXeXeX )](*)([)( 21 jjj
o eXeXeX
)()( * je
je eXeX
)()( * jo
jo eXeX
)()( jeXnx
n
jnjnj
n
eXenxenx )()()(
)()( jeXnx
)()(* * jeXnx)()( jeXnx
n
njnj
n
enxenx*
)()(*
*
)(n
njenx )(* jeX
)()(* * jeXnx)()( jeXnx
)()}(Re{ je eXnx
)()( jeXnx
)()}(Im{ jo eXnxj
)](*)([)}(Re{ 21 nxnxnx
)]()([)](*)([ *
21
21 jj eXeXnxnx
)](*)([)}(Im{ 21 nxnxnxj
)]()([)](*)([ *
21
21 jj eXeXnxnx
)()( jRe eXnx
)()( jeXnx
)()( jIo ejXnx
)](*)([)( 21 nxnxnxe
)]()([)](*)([ *
21
21 jj eXeXnxnx
)](*)([)(21 nxnxnxo
)]()([)](*)([ *
21
21 jj eXeXnxnx
x(n)
)()(* * jeXnx)()( jeXnx
)()( jR
jR eXeX
)()( jI
jI eXeX
|)(||)(| jI
jI eXeX
)()( jj eXeX
)()()()( jj ebYeaXnbynax
11
)()(])()([n
nj
n
njnj
n
enybenxaenbynax
)()( jj ebYeaX
)()( jnjd eXennx d
nj
ndd ennxnnx )()]([
)( jnj eXe d
)()( dnnj
n
enx
nj
n
nj enxe d )(
)()( )( 00 jnj eXnxe
n
njnjnj enxenxe )()]([ 00
n
njenx )( 0)(
)( )( 0jeX
)()( jeXnx
n
njenxnx )()]([
n
njenx )()(
)( jeX
)()( jeXd
djnnx
n
njennxnnx )()]([
n
nj
d
denx
j)(
1
n
njenxd
dj )( )( njeX
d
dj
)()()()()()( jjj
k
eHeXeYknhkxny
n
njenyny )()]([
n
nj
k
eknhkx )()(
k n
njeknhkx )()(
k n
knjenhkx )()()(
k n
njkj enhekx )()(
)()( jj eHeX
deWeXeYnwnxny jjj )()(2
1)()()()( )(
n
njj enxnweY )()()(
n
njnjj edeeXnw )()(2
1
deeXnwn
njj )()()(2
1
denweXn
njj )()()(2
1
deWeX jj )()(2
1 )(
n
jj deYeXnynx )()(2
1)(*)( *
n n
jjj deYeXenynx )()(2
1)(*)( )(
)()(*
)()(* j
j
eYny
eXnx
deWeXeYnwnxny jjj )()(2
1)()()()( )(
n
j deXnx 22 |)(|2
1|)(|
nn
nxnxnx )(*)(|)(| 2
deXeX jj )()(2
1 *
deX j 2|)(|2
1
cc
)( jeH
c
cjeH0
||1)(
deeHnh njjc
c
)(2
1)(
c
c
de nj
2
1
c
c
njdenj
nj )(2
1
c
c
njenj2
1
n
ncsin
-60 -40 -20 0 20 40 60-0.2
0
0.2
0.4
0.6
,2,1,0 sin
)( nn
nnh c
-60 -40 -20 0 20 40 60-0.2
0
0.2
0.4
0.6
,2,1,0 sin
)( nn
nnh c
njM
Mn
cj en
neH
sin)(
-4 -3 -2 -1 0 1 2 3 4-1
0
1
2
M=3
-4 -3 -2 -1 0 1 2 3 4-1
0
1
2
M=5
-4 -3 -2 -1 0 1 2 3 4-1
0
1
2
M=19
njM
Mn
cj en
neH
sin)(
n
njj enxeX )()(
Analysis
deeXnx njj )(2
1)(
Synthesis
n
nx |)(| allfor |)(| jeX
|)(|)( |)(|n
nj
n
njj enxenxeX
n
njenx |||)(|
n
nx |)(|
M
Mn
njjM enxeX )()(
Uniform Convergence
0|)()(|lim jM
j
MeXeX
Mean-Square Convergence
0|)()(|lim 2jM
j
MeXeX
)(n 1
)( dnn dnje
)1|(| )( anuan jae1
1
)(nuk
jk
ae)2(
1
1
)()1( nuan n2)1(
1jae
)1|(| )(sin
)1(sinrnu
nr
p
pn
jjp erer 22cos21
1
n
ncsin
||0
||1)(
c
cjeX
otherwise
Mnnx
0
01)(
2/
)2/sin(
]2/)1(sin[ MjeM
nje 0
k
k)2(2 0
)cos( 0nk
jj keke )]2()2([ 00
