dsp_foehu - lec 04 - discrete-time signals and systems
TRANSCRIPT
In particular, we ll discuss
nnxx )},({
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
nn
u(n)
00
01)(
n
nnu
Fact:
)1()()( nunun
nanx )(
)cos()( 0nAnx
njnnjnnj eaeeenx 000 )()(
x(n) N
NNnxnx allfor )()(
njenx 0)(
)()( 0000 )( Nnxeeeenx njNjNnjnj
kN 20 0
2kN
0
2
|)(| 2n
n
nxE
)}()({ nynxyx
)}()({ nynxyx
)}({ nxx
)()( 0nnxny
k
knkxnx )()()(
)7()3()1()3()( 7213 nananananx
)]([)]([)]()([ 2121 nxbTnxaTnbxnaxT
)()( dnnxny
n
k
kxny )()(
Mk
Mk
knxMM
ny1
)(1
1)(
21
)()( dnnxny
n
k
kxny )()(
Mk
Mk
knxMM
ny1
)(1
1)(
21
2)]([)( nxny
)()()( knkxnxk
)()()( knkxTnyk
)]([)()( knTkxnyk
)()( knhkxk
)(*)()()()( nhnxknhkxnyk
)(*)()()()( nhnxknhkxnyk
)(*)()()()( nxnhknxkhnyk
)(*)()(*)( nxnhnhnx
)()()( Nnununx
00
0)(
n
nanh
n
)()()(*)()( knhkxnhnxnyk
)()()(*)()( knhkxnhnxnyk
)()()(*)()( knhkxnhnxnyk
)()()(*)()( knhkxnhnxnyk
1
1
1
)1(
00 11
1)(
a
aa
a
aaaaany
nnn
n
k
knn
k
kn
11
1
0
1
0 11
1)(
a
aa
a
aaaaany
NnnNn
N
k
knN
k
kn
)()()(*)()( knhkxnhnxnyk
1
1
1
)1(
00 11
1)(
a
aa
a
aaaaany
nnn
n
k
knn
k
kn
11
1
0
1
0 11
1)(
a
aa
a
aaaaany
NnnNn
N
k
knN
k
kn
)()( dnnxny
)()( dnnnh
)()(*)( dd nnxnnnx
Mk
Mk
knxMM
ny1
)(1
1)(
21M
Mk
knMM
nh1
)(1
1)(
21
otherwise
MnMMMnh
01
1
)( 21
21
)()()( nuknhn
k
n
k
kxny )()(
k
khS |)(|
x S < y
kk
khMknxkhny |)(|)()(|)(|
S = x y
0)(0
0)(|)(|
)()(
*
nh
nhnh
nhnx
Skh
khkhkxy
kk |)(|
|)(|)()()0(
2
y(n0) x(n) n n0
h(n)
0for 0)( nnh
h(n)=anu(n) a|<1
1
1|)(|
00 k
k
k aakhS
M
kk
N
kk knxbknya
00
)()(
)()( dnnxny
)()1()( nxnyny
Mk
k
knxM
ny0
)(1
1)(
y(n)
M
kk
N
kk knxbknya
00
)()(
M
k
kN
k
k knxa
bkny
a
any
0 01 0
)()()(
)()( dnnxny )()( dnnnh
Mk
k
knxM
ny0
)(1
1)(
Mk
k
knM
nh0
)(1
1)(
)1()(1
1Mnunu
M
)(*)1()(1
1nuMnn
M
Mk
k
knxM
ny0
)(1
1)(
)(*)1()(1
1)( nuMnnM
nh
1
1
M
