DSP_FOEHU - Lec 04 - Discrete-Time Signals and Systems

2. In particular, we ll discuss

3. nnxx )},({

4. 0 20 40 60 80 100
-10
0
10
t (ms)
0 10 20 30 40 50
-10
0
10
n (samples)
(n)
(n)
00
01
)(
n
n
n

5. u(n)
00
01
)(
n
n
nu
Fact:
)1()()( nunun
n
anx )(

6. )cos()( 0nAnx
njnnjnnj
eaeeenx 000 )(
)(

7. x(n) N
NNnxnx allfor)()(
nj
enx 0
)(
)()( 0000 )(
Nnxeeeenx njNjNnjnj
kN 20 0
2k
N
0
2
|)(| 2
n
n
nxE

8. )}()({ nynxyx
)}()({ nynxyx
)}({ nxx
)()( 0nnxny
k
knkxnx )()()(
)7()3()1()3()( 7213 nananananx

10. )]([)]([)]()([ 2121 nxbTnxaTnbxnaxT
)()( dnnxny
n
k
kxny )()(
Mk
Mk
knx
MM
ny
1
)(
1
1
)(
21

11. )()( dnnxny
n
k
kxny )()(
Mk
Mk
knx
MM
ny
1
)(
1
1
)(
21
2
)]([)( nxny

12. )()()( knkxnx
k
)()()( knkxTny
k
)]([)()( knTkxny
k
)()( knhkx
k

13. )(*)()()()( nhnxknhkxny
k
)(*)()()()( nhnxknhkxny
k
)(*)()()()( nxnhknxkhny
k
)(*)()(*)( nxnhnhnx

15. )()()( Nnununx
00
0
)(
n
na
nh
n
)()()(*)()( knhkxnhnxny
k

16. )()()(*)()( knhkxnhnxny
k
)()()(*)()( knhkxnhnxny
k

17. )()()(*)()( knhkxnhnxny
k
1
1
1
)1(
00 11
1
)(
a
aa
a
a
aaaany
nn
n
n
k
kn
n
k
kn
11
1
0
1
0 11
1
)(
a
aa
a
a
aaaany
NnnN
n
N
k
kn
N
k
kn
)()()(*)()( knhkxnhnxny
k
1
1
1
)1(
00 11
1
)(
a
aa
a
a
aaaany
nn
n
n
k
kn
n
k
kn
11
1
0
1
0 11
1
)(
a
aa
a
a
aaaany
NnnN
n
N
k
kn
N
k
kn

18. )()( dnnxny
)()( dnnnh
)()(*)( dd nnxnnnx

19. Mk
Mk
knx
MM
ny
1
)(
1
1
)(
21
M
Mk
kn
MM
nh
1
)(
1
1
)(
21
otherwise
MnM
MMnh
0
1
1
)( 21
21
)()()( nuknh
n
k
n
k
kxny )()(

20. k
khS |)(|

21. x S < y
kk
khMknxkhny |)(|)()(|)(|
S = x y
0)(0
0)(
|)(|
)(
)(
*
nh
nh
nh
nh
nx
S
kh
kh
khkxy
kk |)(|
|)(|
)()()0(
2

22. y(n0) x(n) n n0
h(n)
0for0)( nnh
h(n)=anu(n) a|<1
1
1
|)(|
00 k
k
k a
akhS

23. M
k
k
N
k
k knxbknya
00
)()(
)()( dnnxny
)()1()( nxnyny
Mk
k
knx
M
ny
0
)(
1
1
)(

24. y(n)
M
k
k
N
k
k knxbknya
00
)()(
M
k
k
N
k
k
knx
a
b
kny
a
a
ny
0 01 0
)()()(
)()( dnnxny )()( dnnnh

25. Mk
k
knx
M
ny
0
)(
1
1
)(
Mk
k
kn
M
nh
0
)(
1
1
)(
)1()(
1
1
Mnunu
M
)(*)1()(
1
1
nuMnn
M
Mk
k
knx
M
ny
0
)(
1
1
)(
)(*)1()(
1
1
)( nuMnn
M
nh
1
1
M