• • • ) ( T ) ( t x a Sampling ) ( ] [ s a nT x n x = • • • 5 t ) ( s T ) ( ] [ s a nT x n x = n 1 − 0 1 2 3 4 • 5 Called; (sequence) 1. Why Sequence? 2. Absolute time is irrelavant Figure 2.1
•• •)(T
)( tx a Sampling )(][ sa nTxnx =
• • • 5
t
)( sT
)(][ sa nTxnx =
n1− 0 1 2 3 4 •5
Called; (sequence)
1. Why Sequence?
2. Absolute time is irrelavant
Figure 2.1
][nδ
11
n01− 1 2 32−
Figure 2.2
][nx
1 ][][ knkx −δ1 ][][ knkx δ
bi h
n01− 1 23
k
][kxx k =magnitude
numberweight ,
Figure 2.3
01− 1 2 3 4 52−3− n
Figure 2.4
0 ω0ω π2 πω 20 + π4 πω 40 + π6Frequency range
Same frequenciesq y g
of interest
Figure 2.5
( )222 ππT
][nx
(sec)210
π==Ω
=T
][1 nx
π3 π5)3(t π
0
1 2 3 4 5
π π2 π4
0 6 7
)4
( nt =
)34( tnπ
=π6
π2=T
38
=N 8=N
3π
3
Figure 2.6
][nx ][2 nx
2 3 4 π3)( nt
• • • • • • • •
0
1 π2π4
0
)( nt =5 6
π2=T
Never repeats itself • • • • • • • •
Figure 2.7
}{Τ
][ny][nx}{•Τ
]}[{][ T ]}[{][ nxTny =
Figure 2.8
}{ •Τ
][ ny][ nx}{
]2[][ nxny =
• • • • • • • •
][ nx
• • • • • • • •
2=M
0 1 2 3 4 5 n• • • • • • • •0 1 2 3 4 5 n
• • • • • • • •
Figure 2.9
][ny][nx
}{Τ
][ny][nxDLTI
}{•Τ
Figure 2.10
][ny][nx
][h
][ny][nxDLTI
][nh
Figure 2.11
][nx ][nh
1 1
n1 0 1 n21
0 11− 0 1
1−
0 1
1−
Figure 2.12
]1[1][ +⋅= nhnh
21
1
][nyn
2
0
][ny
11
]1[)1(][ = nhnh
∑ n
21
−
0 1 22
1−
]1[)1(][ −⋅−= nhnh
n1
0 1 2
21−
21
1−
Good for evaluating y[n] for the cases where x[n] & h[n] are of finite duration
Figure 2.13
][ kx ][
][ kh − ][
k1− 0 1
Figure 2.14
][ ny
11
n10 1 2
21
1− 3 43− 2− • • • • • • • •
• • • • • • • •
• • • • • • • •
• • • • • • • •
2−
1−
Figure 2.15
][ny][nx
}{•Τ
][ny][nx DLTI
][nh
Where ]}[{][ nnh δΤ=
Figure 2.16
][ny][)( i l t ][ny][nx][nh
][nx][1 nh ][2 nh • • • • • • • • ][nhn
][ny )(equivalent
Figure 2.17
][1 nh
][ny][nx][nh][nx
][2 nh
][ny
)(equivalent
∑
][nhn
Figure 2.18
][nh ][nh 01:10:0<<−
<<ax
a
01− 1 22− n 0 1 2 n3 4 5
Figure 2.19
][nx
][δ
][nx
][
][nδ
)(Q
][][ nxny =][nx][
][nδ][nx
][nx)(
][][ nxny][nx][][ nnh δ=][nx
][1 nh
][nyAccumulator Backward
Difference
][2 nh
Figure 2.20
][nx ][nyAccumulator BackwardDifference
][nx
][1 nh ][2 nh
Figure 2.21
][nx∑
][ny
]1[ −ny One Simplep
Delay
Figure 2.22
][ nh
• • • • • • • •• • • • • • • •
samplesM )1( 2 +
0 1 2 3 4 2M n
Figure 2.23
∑][nx
11+M
][1 nx+ ][][ nunh = ][ny∑
Attenuator12 +M
− Accumulator
(M2+1)Simple
Delay
Figure 2.24
][nynjenx ω][
}{•Τ
][nyjenx =][ DLTI
][nh
Figure 2.25
][ny][nx
)( samplesn d
][ny][nx Delay
Figure 2.26
)( ωjLP eH )( ωj
LP eH
ωcω πcω− 0π−
1
cωπ −2cω πcω− 0π−
1
π2 cωπ +2
passband passband
1 )( ωjH
πbω− 0π−
1 )( ωjLP eH
π2 ωaω− aω bωb
passband
a a b
passband
1)( ωj
LP eH
πcω− 0π−
passband
π2 ωcω
passband
1 )( ωjLP eH
πbω− 0π−
1 )(LP eH
π2 ωaω− aω bω
Figure 2.27
stopband stopband
)( ωjeX][ nx
• • • • • • • •
F
][ nx
• • • • • • • •• • • • • • • •
• • • • • • • •
π2
π2− 0 π202 ωπ +
ω2− 1− 0 1 2 3 n π40ω0
Figure 2.28