4장 Sampling of Continuous-Time Signals
• Discrete-Time 신호는 대부분 Continuous-Time 신호를 Sampling해서 얻어진다.
• Sampling은 대부분 Periodic Sampling이다.
• Sampling 에서 Continuous-Time 신호는 Aliasing 최소화를 위해 대역이 제한 ( band limited) 되어 있다고 가정한다.
) nT ( x )
t ( x )
n (
x =
c t nT=
c×
=1
• The Xs(t) is a continuous-time signal that is zero except at integer multiples of T
• The x(n) is indexed on the integer variable n, which in effect introduce a time
sampling with periodic impulse train
4.1 Time and Frequency Domain Representation of Sampling
-2T -T 0 T 2T
3
4.1.1 Sampling
(1) • x(t)
• X(jW)
,
N0 ) j (
X W = W > W
×
å å
¥
-¥
=
¥
-¥
=
W - W p d
= W
×
- d
=
×
k n
) s k T (
) 2 j ( S
) nT t
( )
t ( s ) 2 (
å
å å
¥
-¥
=
¥
-¥
=
¥
-¥
=
W - W
= W
* p W
= W
×
- d
= -
d
=
=
×
k
c c
n n
c c
c s
)) s k (
j ( T x
) 1 j ( S ) j ( 2 x
) 1 j ( Xs
) nT t
( ) nT ( x )
nT t
( )
t ( x ) t ( s ) t ( x ) t ( x ) 3 (
) rate Nyquist
or ( frequency Nyquist
: 2
T 2 s 2
: g
sin Alia )
4 (
N
N
W
×
W p >
= W
× 을 피하기 위해
5
4.1.2 Reconstruction
(1) Ideal LPF ( Reconstruction filter )
( )
, T otherwise 0
) T j ( H
T t
T sin t )
t ( h
C N
S C
N
c r
r
= p W W
- W
<
W
<
W
×
÷ ÷ ø ö ç ç
è
æ W < W
= W
×
p
÷ ø ç ö
è æ p
=
×
) nT ( x ) n m ( ) nT ( ) x
n m (
) n m ( ) sin nT ( x )
mT ( x
T / ) nT t
(
) T / ) nT t
( )sin(
nT ( x )
t ( h ) t ( x ) t ( x ) 2 (
c c
c r
n
c r
s r
= -
d - =
p
-
= p
×
- p
-
= p
*
=
×
å å
å
¥
-¥
=
•The ideal lowpass filter interpolates between the impulses of Xs(t) to construct A continuous-time signal xr(t)
4.1.3 Xc(jW) 와 X(e
jw)
T
) nT ( x ) n ( x ) 4 (
T )) k 2 (T
j ( T Xc
) 1 e ( X
) s k (
j ( T Xc
) 1 e ( X
) e ( X )
e ( X ) j ( Xs )
3 (
e ) n ( x )
e ( X
) nT ( x ) n ( x ) 2 (
e ) nT ( Xc
dt e
) nT t
( ) nT ( Xc dt
e ) t ( Xs )
j ( Xs )
1 (
c k j
k T
j
T j T
j n
n j j
c n
nT j
t j t
j
W
= w
×
=
×
- p
= w
×
W - W
=
×
=
= W
×
=
×
=
×
=
- d
=
= W
å å å å
ò ò
¥
-¥
= w
¥
-¥
= W
W W
= w w
¥
-¥
=
w - w
¥
-¥
=
W -
¥
¥ -
W
¥ -
¥ -
W -
7
• A Frequency scaling, w=nT, is directly associated the fact that there is a time normalization in the transformation from xs(t) to x(n)
4.1.4 Frequency-Domain Description of the Ideal Interpolation
), T j ( X T ))
k 2 (
j ( T X
. 1 T ) j ( X ) 3 (
otherwise
0 T
) T j ( H ) 2 (
) e ( X ) j ( H
e ) j ( H ) n ( x dt
e ) nT t
( h )
n ( x
dt e
) nT t
( h ) n ( x dt
e ) t ( x )
j ( X ) 1 (
c c
r r
T j r
nT j n
r t
j n
r
t j n
r t
j r
r
£ p W W p =
- W
= W
ç ç è
æ p
£
= W W
W
=
W
= -
=
-
=
= W
å
å å ò
ò å ò
W
W -
¥
-¥
= W
-
¥
-¥
=
¥
¥ -
W
¥ -
¥ -
¥
-¥
=
¥
¥ -
W -
(4) The ideal LPF Hr(jW) selects the base period of the resulting periodic
