4.4 Changing the Sampling Rate
4.4.1 Sampling Rate Reduction by an Integer Factor
compressor(discrete-time samples)
• sampling rate compressor or down sampling or decimation
• sampling rate가 1/M로 줄게 되므로 (T’=MT) xc(t)는 로 대역 제한 신호이어야
down sample후 aliasing이 없다.
MT
< p W
(1) FT of x
d(n)
• xd(n) = x(nM) = xc(nMT)
) Z ( x Z
) k ( x Z
) nM ( x )
z ( x
) nM ( x ) nM ( x ) n ( x
otherwise 0
M of multiple is
n ) n ( ) x
n ( x
Z ) nM ( x Z
) n ( x )
z ( x
M 1 1 k
M k 1
n
n 1
d
1 d
1
n
n n
n d
d
=
=
=
×
=
=
×
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= æ
×
=
=
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å å
å å
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=
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=
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( )
å
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å å å
å å
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= w -
- -
= -
=
-
=
- p
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=
p - - -
=
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- p
p w
p
=
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÷ =
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= æ
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×
1 M
0 k
) (
j j d
1 jk M
0 k 1
d
1 M
0 k
1 M
0 k
M jk2
n
n M jk2 1
M
0 k n
M n n jk2 1
M k 2 M
M 2 M
1 M
1
e M X
) 1 e ( x
) e
Z ( M X
) 1 Z ( x ) z ( x
) Ze
( M X
Ze 1 ) n ( M x
1
Z e
) n ( M x
) 1 z ( x
å 이므로
-
=
=
pçç è
= æ
=
×
1 M
0 k
jk M
M
M 1
M n
e
2M ) 1
n ( C
otherwise ,
0
M of multiple is
n ,
) 1 n ( C
) n ( x ) n ( C )
n
(
x
(2) Graphical illustration of x
d(e
jw), M=3
frequency-domain effect of decimation with M=3
(3) Decimation에서 Aliasing을 줄이기 위해서는 prefiltering이 필요하다.
general system for sampling rate reduction by M
4.4.2 Increasing the Sampling Rate by an Integer Factor
general system for sampling rate increase by L
expender and interpolation
(1) x
i(n) 과 x
e(n)
( )
) e ( X e
) k ( x
e ) kL n
( ) k ( x )
e ( X
) kL n
( ) k ( x )
n ( x
otherwise 0
L 2 , L , 0 n ) x
n ( x
L 2 , L , 0 n ), ( x ) ( x ) n ( x
L j k
kL j n
n j k
j e
k e
L n e
L nT L c
n i
w
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-¥
=
w -
¥
-¥
=
w -
¥
-¥
= w
¥
-¥
=
å
å å å
=
=
÷ ø ç ö
è
æ d -
=
×
- d
=
×
÷÷ ø ö çç è
æ = ± ±
=
×
±
±
=
=
=
×
• Sampling rate expander or upsampling or interpolation
• Sampling rate가 L배 증가한다.
• Interpolation을 위해 gain L, 차단주파수 p/L인 ideal LPF가 필요하다.
• 이득이 L 이고, 차단주파수 p/L인 ideal LPF
(2) Ideal interpolation of x
e(n)
L n
L n n
sin i
L sin )
n (
h
Lnp p
p
=
=
p( )
( )
( )
Ln c( )
nTLi
L i n i
L ) L n (
L L n i
k
i k
i e
i e
i
x x
) n ( x
, L 2 , L n
, 0 ) n ( h , )
( 0 L n
; 1 ) 0 ( h ,
) sin ( x )
L n
( h ) ( x
) k n ( h ) L k
( ) ( x
) k n ( h ) k ( x )
n ( h ) n ( x ) n ( x
=
=
±
±
=
=
=
= -
=
= p -
=
- -
d
=
-
=
*
=
×
å å
å å
å
¥
-¥
=
- p
¥ -
-¥
=
¥
-¥
=
¥
-¥
=
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-¥
=
이므로 이고
그런데 l l L
l l
l
l l
l
l l
l l
(3) Linear interpolation
2
2 2 L j
k k
e e
L n e
sin sin L ) 1 e ( H
) kL n ( h ) k ( x )
k n ( h ) k ( x )
n ( x
otherwise 0
L n ) 1
n ( h
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= é
×
-
= -
=
×
÷ ÷ ø ö ç ç
è
æ - £
=
×
w w w
¥
-¥
=
¥
-¥
=
å
å
l
l l
Linear interpolation by filtering 10
4.4.3 Changing the Sampling Rate by a Noninteger Factor
• T’ = TM / L
• M > L : increasing in the sampling period ( decreasing in the sampling rate )
• M < L : opposite of the above
• Since the interpolation and decimation filter are in cascade, they can be combined into system for changing the sampling rate by a noninteger factor
4.5 Multirate Signal Processing
4.6 Digital Processing of Analog Signal 4.6.1 Prefiltering to Avoid Aliasing
practical digital processing of analog signals
4.6.2 A/D Conversion
4.6.3 Analysis of Quantization Errors ) e ( H ) j ( H ) j ( H
LPF ideal
) j ( H
0 ) e ( ) H
j ( H
0 ) 1
j ( H
T j aa
eff aa
C C T
j eff
C C T aa
W W
p
W
@ W W
×
÷ ÷ ø ö ç ç
è æ
W
>
W
W
<
= W W
×
÷ ÷ ø ö ç ç
è æ
W
>
W
<
W
<
= W W
×
아니므로 가
가
(1) Ideal reconstruction filter
ç ç è æ
>
W
<
= W
W
pp
T T
, 0
, ) T
j ( Hr
(2) D/A converter
2
j T
2 T 0
0
) e sin(
) 2 j ( H
otherwise 0
T t 0 ) 1
t ( h
W
= W W
çç è
æ < <
=
W
ç ç ç ç
è æ
> p W
< p W W =
= W W
W
×
W W
W
, T 0
, T ) e
sin(
) j ( H
) j ( ) H
j ( H
) j ( H
converter A
/ D
2 T j
2 T 2
T
0 r r
^
r
필터
^위한 보상하기
를
4.6.4 D/A Conversion
frequency response of D/A converter and ideal compensated reconstruction filter
