Chapter 4 sampling of continous-time signals

4.5 changing the sampling rate using discrete-time processing

4.1 periodic sampling

4.2 discrete-time processing of continuous-time signals

4.3 continuous-time processing of discrete-time signal

4.4 digital processing of analog signals

4.1 periodic sampling

1.ideal sample

) ( | ) ( ] [nT x t x n xc nT t c

T ： sample periodfs=1/T:sample rateΩs=2π/T:sample rate

Figure 4.1 ideal continous-time-to-discrete-time(C/D)converter

Figure 4.2(a) mathematic model for ideal C/D

n

nTt )(

time normalization tt/T=n

Figure 4.3

NNs

frequency spectrum change of ideal sample

NNs

aliasing frequency

No aliasing

aliasing

k

scs kjXT

jX ))((1

)(

T

k

c

Tsj

TkjXT

jXeX

)/)2((1

|)()( /

2/s

2

) 1. 0 cos( ) 1. 2 cos(n n Period =2πin time domain ：

w=2.1πand w=0.1πare the same

trigonometric function property

) 9. 0 cos( ) 1. 1 cos(n n high frequency is changed into low frequency in time domain ：

w=1.1π and w=0.9πare the same

trigonometric function property

2.ideal reconstruction

Figure 4.10(b) ideal D/C converter

Figure 4.4

ideal reconstruction in frequency domain

2/ s

Figure 4.5EXAMPLE Take sinusoidal signal for example to understand aliasing from frequency domain

0 s

Hzftttxa 5,10),52cos()(

Hzf 358'

Reconstruct frequency:

EXAMPLE

Sampling frequency:8Hz

Figure 4.10(a) mathematic model for ideal D/C

c

cr

crsr

TjH

jXjHjXjX

||

||

0)(

)()()()(

ideal reconstruction in time domain

)(sin xc

Tt

Tt

Tt

t

dTedejHjHIFTth

jHjXjX

c

tjtjrrr

rsr

C

C

/

)/sin(

/

)sin(2

1)(

2

1)}({)(

)()()(

x

xxc

TnTt

TnTtnx

Tt

TtnTtnx

Tt

TtnTtnxthtxtx

nn

nrsr

)sin()(sin

/)(

]/)(sin[][]

/

)/sin()(][[

/

)/sin(])(][[)()()(

Figure 4.9

EXAMPLE

EXAMPLE

understand aliasing from time-domain interpolation

3.Nyquist sampling theorems

NNs

NNs

Nc

c

jX

withsignalitedbandabetxlet

||,0)(

lim)(

ingundersampl

ngoversampli

ratenyquist

frequencynyquist

Ns

Ns

N

s

:2

:2

:2

:2/

,2,1,0),(][

mindet)(

nnTxnx

samplesitsbyederuniquelyistxthen

c

c

NsNs

NNs

fT

forT

isthatif

2)1

(,2)2

(

,

Nyquist sampling theorems:

Figure 4.41 Digital processing of analog signals

2/sc

)( jHaa

examples of sampling theorem（ 1 ）

8kHz

The highest frequency of analog signal ,which wav file with sampling rate 16kHz can show ， is ：

The higher sampling rate of audio files, the better fidelity.

according to what you know about the sampling rate of MP3 file ， judge the sound we can feel

frequency range （ ）

（ A ） 20~44.1kHz （ B ） 20~20kHz （ C ） 20~4kHz （ D ） 20~8kHz

B

examples of sampling theorem（ 2 ）

:

)cos()10cos()(][

)1.0(10

5,10),10cos()(

signaltionreconstrucdraw

nnTnTxnxdraw

sTHzf

Hzftttx

a

a

s

n TnTt

TnTtnxty

/)(

]/)(sin[][)(

EXAMPLE Matlab codes to realize interpolation

T=0.1; n=0:10; x=cos(10*pi*n*T); stem(n,x);dt=0.001; t=ones(11,1)* [0:dt:1]; n=n'*ones(1,1/dt+1);y=x*sinc((t-n*T)/T); hold on; plot(t/T,y,'r')

Supplement: band-pass sampling theorem ：

Nff

fM

ff

fN

NMfff

LH

H

LH

H

LHs

int

)/1)((2

BfH 5

4fs

BfB H 54

)/1(2)(

2/

2/2

2

NMBN

NMB

N

BfBNff

fNf

HHs

Hs

4.1 summary

1.representation in time domain of sampling

) ( | ) ( ] [nT x t x n xc nT t c

2.changes in frequency domain caused by sampling

k

cj

k

scs

TkjXT

eX

kjXT

jX

)/)2((1

)(

))((1

)(

3. understand reconstruction in frequency domain

4. understand reconstruction in time domain

5. sampling theorem

NNs

NNs

Requirements and difficulties ：frequency spectrum chart of sampling and reconstructioncomprehension and application of sampling theorem

4.2 discrete-time processing of continuous-time signals

Figure 4.11

Figure 4.12

EXAMPLE

T

TeHjHTj

eff/||0

/||)()(

conditions ： LTI ； no aliasing or aliasing occurred outside the pass band of filters

Figure 4.13

EXAMPLE

aliasing occurred outside the pass band of digital

filters satisfies the equivalent relation of frequency response mentioned before.

4.3 continuous-time processing of discrete-time signal

Figure 4.16

||)/()( forTjHeH cj

Figure 4.12

c

jj eeH )(

EXAMPLE Ideal delay system ： noninteger delay

4.4 digital processing of analog signals

Figure 4.41

quantization and coding

Figure 4.46(b)

Sampling and holding

Figure 4.48

uniform quantization and

coding

)(25.16

numbersbit :,2/2

dBBSNR

BX Bm

Figure 4.51

quantization error of 3BIT

quantization error of 8BIT

0 量化前的电平

量化后的电平

nonuniform quantization

x4

3 32

3

1

32

2 21

34

34

1

1 1

3

4

码书

codeword

c0

codeword c1

codeword c2

codeword c3

index0

d(x,c0)=5

d(x,c1)=11

d(x,c2)=8

d(x,c3)=8

一维信号例子：

x

vector quantization

Example: image coding

Initial image block（ 4 gray-levels， dimentions k=4×4=16）

x 0 1 2 3

Code book C ＝｛ y0, y1 , y2, y3｝

y0 y1 y2 y3

codeword y1 is the most adjacent to x， so it is coded by the index “01”.

d(x,y0)=25

d(x,y1)=5

d(x,y2)=25

d(x,y3)=46

vector quantization

Figure 4.53 D/A 过程

)()( tRth N

reconstruction

Figure 4.5

record the digital sound

Influence caused by sampling rate and quantizing bits

Different tones require different sampling

rates.

4.1~4.4 summary

1. representation in time domain and changes in frequency domain of sampling and reconstruction. sampling theorem educed from aliasing in frequency domain

2. analog signal processing in digital system or digital signal in analog system , to explain some digital systems ， their frequency responses are linear in dominant period

3. steps in A/D conversion

Requirements and difficulties ：sampling processing in time and frequency domain ， freq

uency spectrum chart;comprehension and application of sampling theorem;frequency response in discrete-time processing system of c

ontinuous-time signals;

4.5 changing the sampling rate using discrete-time processing

4.5.1 sampling rate reduction by an integer factor(downsampling, decimation)

4.5.2 increasing the sampling rate by an integer factor(upsampling, interpolation)

4.5.3 changing the sampling rate by a noninteger fact

4.5.4 application of multirate signal processing

4.5.1 sampling rate reduction by an integer factor(downsampling, decimation)

:

][][

compressorratesamplinga

nMxnxd

Figure 4.20

time-domain of downsampling ： decrease the data ， reduce the sampling rate

frequency-domain of downsampling ： take aliasing into consideration

1

0

/)2(/)2( 1 M

i

MijMijd eX

MeX

M=2 ， fs’=fs/M,T’=MT

)2/)2(2/

2

1)( jjj

d eXeXeX

)( jdeX

)( jeX

202

202

2/1

Figure 4.21(c)(d)

EXAMPLE M=2

)3/)4()3/)2(3/

3

1)( jjjj

d eXeXeXeX

)(jeX

)(j

d eX 202

202

EXAMPLE M=3

Figure 4.22(b)(c)

EXAMPLE M=3 ， aliasing

frequency spectrum after decimation ： period=2π ，M times wider ， 1/M times higher

Figure 4.23

MN / Condition to avoid aliasing ：

Total downsampling system ： Total system

4.5.2 increasing the sampling rate by an integer factor(upsampling, interpolation)

:exp

][][][,

2,,0

0

]/[][

anderratesamplinga

kLnkxnxor

other

LLnLnxnx

k

e

e

][nxe][nx L

L=2 ， fs’=Lfs,T’=T/L

time-domain of upsampling ： increase the data ， raise the sampling rate

frequency-domain of upsampling ： need not take aliasing into consideration

)()( jLje eXeX

Figure 4.25

EXAMPLE

L=2

transverse axis is 1/L timer shorter ， magnitude has no change. L mirror images in a period. Period=2π ， also period =2 π /L

Take T’=T/L as D/C:

NTT

LL

'/

frequency domain of reverse mirror-image filter

total upsampling system ： total system

Figure 4.24

kki

ik

ik

iei

ji

LkLnL

kLn

kxkLnhkx

nhkLnkx

nhkLnkxnhnxnx

n

LnLeHIFTnh

)(

)(sin

][][][

])[][]([

][)][][(][][][

/sin][][

time-domain explanation of reverse mirror-image filter :slowly-changed signal by interpolation

EXAMPLEtime-domain process of

mirror-image filter

Figure 4.27

Use linear interpolation actually

4.5.3 changing the sampling rate by a noninteger factor

Figure 4.28

M

Lff ss '

4,3

300'400

ML

HztosignalsHzchangeEXAMPLE

)( jeX

202)(),( jeiXjeeX

24/02

202

)( jeY

Advantages of decimation after interpolation ：1.Combine antialiasing and reverse mirror-image filter2.Lossless information for upsampling

4.5.4 application of multirate signal processing

Figure 4.43

1.Sampling system ： replace high-powered analog antialiasing filter and low sampling rate with low-powered analog antialiasing filter , oversampling and high-powered digital antialiasing filter, decimation. Transfer the difficulty of the realization of high-powered analog filter to the design of high-powered digital filter.

FIGURE 4.44

x(t) 模拟重构滤波 x[n] 反镜像滤波 零 阶 保 持h1(t)

↑ L

xe[n] xi[n]

2.reconstruction system ： replace high-powered analog reconstructing filter with interpolation, high-powered digital reverse mirror-image filter and low-powered analog analog reconstructing filter.

][0 ny][ nx ][0 nh

][1 nh

][ nh N ][ nh N

][0 nh

][1 nh

M

M

M M

M

M ][ nx

][1 ny

][ ny N

.........................................

analysis and synthesis of sub band

3. filter bank

In MP3, M=32 ， sub-band analysis filter bank is 32 equi-band filters with center frequency uniformly distributed from 0 to π ：

MP3 coders use different quantization to realize compression for signals yi[n] in different sub-bands.

20

)(1jeY

20

)( jeX

20

)('0jeY

20

)('1jeY

20

)(0jeY

20

)(''0jeY

20

)(''1jeY

20

)('0jeY

20

)('1jeY

example ： compression for M=2

fsbitfsbitfsbit

fbit s

122/82/16ncompressioafter ratebit

16ncompressio before ratebit

：：

decimation an d interpolation to realize pitch scale

4.pitch scale ： decimation or interpolation ， sampling rate of reconstruction is not changed.

4.5 summary

requirement ：frequency spectrum chart of interpolation and decimation

4.5.1 sampling rate reduction by an integer factor4.5.2 increasing the sampling rate by an integer factor4.5.3 changing the sampling rate by a noninteger fact

4.5.4 application of multirate signal processing

exercises

4.15 (b)(c)4.24(a)(b)4.26 only for ωh= π /4