DSP_FOEHU - Lec 05 - Frequency-Domain Representation of Discrete Time Signals

1. nj
enx )(
k
knxkhny )()()(
k
knj
ekh )(
)(
jn
k
jk
eekh )(
jnj
eeH )(

2. nj
e
jnj
eeH )()( j
eH
k
jkj
ekheH )()(
k
jkj
ekheH )()(
)()()( jj
R
j
ejHeHeH
)(
|)(|)(
j
eHjj
eeHeH

3. )()( dnnxny )()( dnnnh
dnj
k
nj
d
k
njj
eenkekheH )()()(
1|)(| j
eH
d
j
neH )(
dnjj
eeH )(
)cos()( 0nAnx
njjnjj
ee
A
ee
A
nx 00
22
)(
dd njnjjnjnjj
eee
A
eee
A
ny 0000
22
)(
)()( 00
22
dd nnjjnnjj
ee
A
ee
A
])(cos[)( 0 dnnAny

4. k
jkj
ekheH )()(
k
jkj
ekheH )2()2(
)()(
k
jk
ekh )(
)( j
eH
,2,1,0
)()( )2(
m
eHeH mjj
k
jkj
ekheH )()(
,2,1,0
)()( )2(
m
eHeH mjj
|)(| j
eH

5. k
jkj
ekheH )()(
,2,1,0
)()( )2(
m
eHeH mjj
|)(| j
eH
k
jkj
ekheH )()(
,2,1,0
)()( )2(
m
eHeH mjj
|)(| j
eH

6. |)(| j
eH
|)(| j
eH
|)(| j
eH
Mk
k
knx
M
ny
0
)(
1
1
)(
M
k
kn
M
nh
0
)(
1
1
)(
k
jkj
ekheH )()(
M
k
jk
e
M 01
1
j
Mj
e
e
M 1
1
1
1 )1(
)(
)(
1
1
2/2/2/
2/)1(2/)1(2/)1(
jjj
MjMjMj
eee
eee
M
)(
)(
1
1
2/2/
2/)1(2/)1(
2/
jj
MjMj
Mj
ee
ee
e
M
)2/sin(
]2/)1(sin[
1
1 2/ M
e
M
Mj

7. )2/sin(
]2/)1(sin[
1
1
)( 2/ M
e
M
eH Mjj
)2/sin(
]2/)1(sin[
1
1
|)(|
M
M
eH j

8. 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 nj
m
m
mj
)(
2
1
deemx njmj
m
m
)(
2
1
demx mnj
m
m
)(
)(
2
1
2)(
de mnj
deeXnx njj
)(
2
1
)(

9. )(
)(
1 )(
mndje
mnj
mnj
)(
)(
1 mnj
e
mnj
)()(
)(
1 mnjmnj
ee
mnj
)(sin2
)(
1
mnj
mnj
0
)(
)(sin2
mn
mn
de mnj )(
n
n
njj
enxeX )()(
deeXnx njj
)(
2
1
)(
deemx nj
m
m
mj
)(
2
1
deemx njmj
m
m
)(
2
1
demx mnj
m
m
)(
)(
2
1
deeXnx njj
)(
2
1
)(
n
n
njj
enxeX )()(
deeXnx njj
)(
2
1
)(
deemx nj
m
m
mj
)(
2
1
deemx njmj
m
m
)(
2
1
demx mnj
m
m
)(
)(
2
1
deeXnx njj
)(
2
1
)(

10. n
n
njj
enxeX )()(
Analysis
deeXnx njj
)(
2
1
)(
Synthesis Inverse Fourier Transform
(IFT)
Fourier Transform
(FT)
)]([)( nxeX j
)]([)( 1 j-
eXnx
)()( j
eXnx
n
n
njj
enxeX ][)(
Fourier Transform (FT)
)()()( j
I
j
R
j
ejXeXeX

11. )(
|)(|)(
j
eXjjj
eeXeX
)()()( j
I
j
R
j
ejXeXeX
)( j
eX
)( j
R eX
)( j
I eX
|)(| j
eX
)( j
eX

12. )()( *
nxnx ee
)()( *
nxnx oo
)()()( nxnxnx oe
)](*)([)( 2
1
nxnxnxe )](*)([)( 2
1
nxnxnxo

13. )()()( j
o
j
e
j
eXeXeX
)](*)([)( 2
1 jjj
e eXeXeX )](*)([)( 2
1 jjj
o eXeXeX
)()( * j
e
j
e eXeX
)()( * j
o
j
o eXeX

14. )()( j
eXnx
n
jnjnj
n
eXenxenx )()()(
)()( j
eXnx
)()(* * j
eXnx)()( j
eXnx
n
njnj
n
enxenx
*
)()(*
*
)(
n
nj
enx )(* j
eX

15. )()(* * j
eXnx)()( j
eXnx
)()}(Re{ j
e eXnx
)()( j
eXnx
)()}(Im{ j
o eXnxj
)](*)([)}(Re{ 2
1
nxnxnx
)]()([)](*)([ *
2
1
2
1 jj
eXeXnxnx
)](*)([)}(Im{ 2
1
nxnxnxj
)]()([)](*)([ *
2
1
2
1 jj
eXeXnxnx

16. )()( j
Re eXnx
)()( j
eXnx
)()( j
Io ejXnx
)](*)([)( 2
1
nxnxnxe
)]()([)](*)([ *
2
1
2
1 jj
eXeXnxnx
)](*)([)( 2
1
nxnxnxo
)]()([)](*)([ *
2
1
2
1 jj
eXeXnxnx
x(n)
)()(* * j
eXnx)()( j
eXnx
)()( j
R
j
R eXeX
)()( j
I
j
I eXeX
|)(||)(| j
I
j
I eXeX
)()( jj
eXeX

17. )()()()( jj
ebYeaXnbynax
11
)()(])()([
n
nj
n
njnj
n
enybenxaenbynax
)()( jj
ebYeaX

18. )()( jnj
d eXennx d
nj
n
dd ennxnnx )()]([
)( jnj
eXe d
)(
)( dnnj
n
enx
nj
n
nj
enxe d
)(
)()( )( 00 jnj
eXnxe
n
njnjnj
enxenxe )()]([ 00
n
nj
enx )( 0
)(
)( )( 0j
eX

19. )()( j
eXnx
n
nj
enxnx )()]([
n
nj
enx )(
)(
)( j
eX
)()( j
eX
d
d
jnnx
n
nj
ennxnnx )()]([
n
nj
d
de
nx
j
)(
1
n
nj
enx
d
d
j )( )( nj
eX
d
d
j

20. )()()()()()( jjj
k
eHeXeYknhkxny
n
nj
enyny )()]([
n
nj
k
eknhkx )()(
k n
nj
eknhkx )()(
k n
knj
enhkx )(
)()(
k n
njkj
enhekx )()(
)()( jj
eHeX
deWeXeYnwnxny jjj
)()(
2
1
)()()()( )(
n
njj
enxnweY )()()(
n
njnjj
edeeXnw )()(
2
1
deeXnw
n
njj )(
)()(
2
1
denweX
n
njj )(
)()(
2
1
deWeX jj
)()(
2
1 )(

21. 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

22. cc
)( j
eH
c
cj
eH
0
||1
)(
deeHnh njjc
c
)(
2
1
)(
c
c
de nj
2
1
c
c
njde
nj
nj
)(
2
1
c
c
nj
e
nj2
1
n
ncsin
-60 -40 -20 0 20 40 60
-0.2
0
0.2
0.4
0.6
,2,1,0
sin
)( n
n
n
nh c

23. -60 -40 -20 0 20 40 60
-0.2
0
0.2
0.4
0.6
,2,1,0
sin
)( n
n
n
nh c
nj
M
Mn
cj
e
n
n
eH
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
nj
M
Mn
cj
e
n
n
eH
sin
)(

24. n
njj
enxeX )()(
Analysis
deeXnx njj
)(
2
1
)(
Synthesis

25. n
nx |)(| allfor|)(| j
eX
|)(|)(|)(|
n
nj
n
njj
enxenxeX
n
nj
enx |||)(|
n
nx |)(|
M
Mn
njj
M enxeX )()(
Uniform Convergence
0|)()(|lim j
M
j
M
eXeX
Mean-Square Convergence
0|)()(|lim 2j
M
j
M
eXeX

26. )(n 1
)( dnn dnj
e
)1|(|)( anuan
j
ae1
1
)(nu
k
j
k
ae
)2(
1
1
)()1( nuan n
2
)1(
1
j
ae

27. )1|(|)(
sin
)1(sin
rnu
nr
p
p
n
jj
p erer 22
cos21
1
n
ncsin
||0
||1
)(
c
cj
eX
otherwise
Mn
nx
0
01
)(
2/
)2/sin(
]2/)1(sin[ Mj
e
M
nj
e 0
k
k)2(2 0
)cos( 0n
k
jj
keke )]2()2([ 00