•• •)(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