Copyright © 2005. Shi Ping CUC Chapter 5 Sampling Rate Conversion Conte nt Introduction Decimation by a factor D Interpolation by a factor I Sampling rate conversion by a rational factor I/D
Dec 23, 2015
Copyright © 2005. Shi Ping CUC
Chapter 5 Sampling Rate Conversion
Content
Introduction
Decimation by a factor D
Interpolation by a factor I
Sampling rate conversion by a rational factor I/D
Copyright © 2005. Shi Ping CUC
In many practical applications of DSP, one is faced with the problem of changing the sampling rate of a signal.
The process of converting a signal from a given rate to a different rate is called sampling rate conversion.
Systems that employ multiple sampling rates in the processing of digital signals are called multirate digital signal processing systems.
Introduction
Copyright © 2005. Shi Ping CUC
There are two general methods to accomplish the sampling rate conversion of a digital signal.
To pass the digital signal through a D/A converter, filter it if necessary, and then to resample the resulting analog signal at the desired rate;
To perform the sampling rate conversion entirely in the digital domain.
The process of reducing the sampling rate by an integer factor D (downsampling by D) is called decimation.
The process of increasing the sampling rate by an integer factor I (upsampling by I) is called interpolation.
Introduction
return
Copyright © 2005. Shi Ping CUC
In downsampling by an integer factor D>1, every D-th samples of the input sequence are kept and others are removed:
)()( Dnxnxd
D)(nx )(nxd
Decimation by a factor D
sfD
f s
Copyright © 2005. Shi Ping CUC
Relationship in time domain
)()()( npnxnx p
)(nx Input sequence
k
kDnnp )()( Periodic train of impulses
)()()( DnxDnxnx pd Output sequence
demo
Decimation by a factor D
Copyright © 2005. Shi Ping CUC
Relationship in frequency domain
2
0
)( )()(2
1)( deXePeX jjj
p
1
0
21
0
2
)()( ,)(1
)(D
n
knD
jD
k
knD
jenpkPekP
Dnp
1)()()(1
0
21
0
2
D
n
knD
jD
n
knD
j
i
eneiDnkP
Decimation by a factor D
Copyright © 2005. Shi Ping CUC
1
0
)( )(1
)(D
k
kjjp
seXD
eX
)()()(
)()()(
Dj
pn
D
nj
pmDn
D
nj
p
m
mjp
m
mjd
jd
eXenxenx
emDxemxeX
Ds
2
1
0
1
0
2
1
0
2
)2
(21
)(1
)()(
D
k
D
kn
njkn
Dj
n
njD
k
knD
j
n
njj
kDD
eeD
eekPD
enpeP
mDnlet
demo
Copyright © 2005. Shi Ping CUC
h(n))(nx )(nxd D
)(' nx
otherwise ,0
|0 ,1)( Dπ|ωeH j
Using a digital low-pass filter to prevent aliasing
Decimation by a factor D
return
Copyright © 2005. Shi Ping CUC
In up-sampling by an integer factor I >1, I -1 equidistant zeros-valued samples are inserted between each two consecutive samples of the input sequence. Then a digital low-pass filter is applied.
otherwise ,0
2 , ,0 ),()( IInI
nxnx p
I)(nx )(nxIh(n))(nx p
demo
Interpolation by a factor I
sf sIf
Copyright © 2005. Shi Ping CUC
Relationship in frequency domain
k
p kInkxnx )()()( )(nx Input sequence
)()(
)()()(
Ij
k
Ikj
n
nj
k
jp
eXekx
ekInkxeX
otherwise ,0
|0 ,)( Iπ|ωIeH j demo
Interpolation by a factor I
return
Copyright © 2005. Shi Ping CUC
If is a rational numberD
IR
h1(n)
h2(n)
I D)(nx
interpolation decimationsf
)(nxI
sIf
)(nxId
sfD
I
Sampling rate conversion by a rational factor I/D
Sampling period
T I
T
I
T
I
T
I
DT
Copyright © 2005. Shi Ping CUC
h (n) I D)(nx )(nxId
otherwise ,0
),min(|0 ,)( Dπ
Iπ|ωIeH j
example1
Sampling Rate Conversion
example2
return
Copyright © 2005. Shi Ping CUC
-15 -10 -5 0 5 10 150
2
n
-15 -10 -5 0 5 10 150
0.5
1
n
-15 -10 -5 0 5 10 150
2
n
-15 -10 -5 0 5 10 150
2
n
)(nx
)(np
)(nx p
)(nxd
return
4D
Copyright © 2005. Shi Ping CUCreturn
hh
)( jeX
0 22
)( jeP
0 22 ss s3s3
D
2
0 22 hh
D
1
ss s3s3
)( jp eX
0 22 hDhD
D
1)( jd eX
Copyright © 2005. Shi Ping CUC
0 4 8 12 16 20 24 28 32 36 40 44 480
2
4
n
0 4 8 12 16 20 24 28 32 36 40 44 480
2
4
n
0 4 8 12 16 20 24 28 32 36 40 44 480
2
4
n
)(nx
)(nx p
)(nxI
return
4I
Copyright © 2005. Shi Ping CUCreturn
7
2
)( jeX
0 22 7
2
7
127
167
12
7
16
7
)( jI eX
0 22 7
2I
)( jId eX
0 22
7 ,2 DI
Copyright © 2005. Shi Ping CUC
P137, No.26 in textbook
,3 ,1 ,0
,4 ,2 ,0 ),()(
n
nnxnx p
)2()( nxnxd
DeX
DeX s
D
k
kjjp
s 2
,)(1
)(1
0
)(
)()( Dj
pj
d eXeX
Copyright © 2005. Shi Ping CUC
4
3
)( jeX
0 22 4
3
4
54
114
5
4
11
1
)( jp eX
04
3 22 4
3
4
54
114
5
4
11
2
1 Ds
2
2
1
)( jp eX
04
3 22 4
3
4
54
114
5
4
11