Chap5-Sampling Rate Conversion

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


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


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


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


sfD

f s


Relationship in time domain

)()()( npnxnx p

)(nx Input sequence

k

kDnnp )()( Periodic train of impulses

)()()( DnxDnxnx pd Output sequence

demo



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



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


h(n))(nx )(nxd D

)(' nx

otherwise ,0

|0 ,1)( Dπ|ωeH j

Using a digital low-pass filter to prevent aliasing


return


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


sf sIf


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


return


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


h (n) I D)(nx )(nxId

otherwise ,0

),min(|0 ,)( Dπ

Iπ|ωIeH j

example1

Sampling Rate Conversion

example2

return


-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


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


0 22 hh

)( jeX

0 22 I

hI

h

)( jI eX

0 22 I

hI

2I

6

)( jp eX

Ih


7

2

)( jeX

0 22 7

2

7

127

167

12

7

16

7

6

)( jd eX

0 22 7

6

3D


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


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


n

)(nx

0

n

)(nx p

0

n

)(nxd

0

2D


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


4

3

)( jeX

0 22 4

3

4

54

114

5

4

11

1

2

1

)( jd eX

0 22 4

62

2

4

6

2

1

)( jd eX

0 22 4

62

2

4

6

return

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Chap5-Sampling Rate Conversion