The document discusses a signal (u(n)) and its relationship to another signal (y(n)) through a linear time-invariant system and convolution. It provides equations to describe the output signal (y(n)) in terms of the input signal (u(n)) using concepts such as impulse response (h(n)) and convolution.

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