Chapter 4 sampling of continous-time signals 4.5 changing the sampling rate using discrete-time...
-
Upload
marilyn-vercoe -
Category
Documents
-
view
223 -
download
0
Transcript of Chapter 4 sampling of continous-time signals 4.5 changing the sampling rate using discrete-time...
![Page 1: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/1.jpg)
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
![Page 2: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/2.jpg)
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
![Page 3: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/3.jpg)
Figure 4.1 ideal continous-time-to-discrete-time(C/D)converter
![Page 4: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/4.jpg)
Figure 4.2(a) mathematic model for ideal C/D
n
nTt )(
time normalization tt/T=n
![Page 5: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/5.jpg)
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
![Page 6: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/6.jpg)
) 1. 0 cos( ) 1. 2 cos(n n Period =2πin time domain :
w=2.1πand w=0.1πare the same
trigonometric function property
![Page 7: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/7.jpg)
) 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
![Page 8: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/8.jpg)
2.ideal reconstruction
Figure 4.10(b) ideal D/C converter
![Page 9: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/9.jpg)
Figure 4.4
ideal reconstruction in frequency domain
2/ s
![Page 10: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/10.jpg)
Figure 4.5EXAMPLE Take sinusoidal signal for example to understand aliasing from frequency domain
0 s
![Page 11: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/11.jpg)
Hzftttxa 5,10),52cos()(
Hzf 358'
Reconstruct frequency:
EXAMPLE
Sampling frequency:8Hz
![Page 12: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/12.jpg)
Figure 4.10(a) mathematic model for ideal D/C
c
cr
crsr
TjH
jXjHjXjX
||
||
0)(
)()()()(
![Page 13: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/13.jpg)
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(])(][[)()()(
![Page 14: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/14.jpg)
Figure 4.9
EXAMPLE
![Page 15: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/15.jpg)
EXAMPLE
understand aliasing from time-domain interpolation
![Page 16: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/16.jpg)
3.Nyquist sampling theorems
NNs
NNs
![Page 17: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/17.jpg)
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:
![Page 18: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/18.jpg)
Figure 4.41 Digital processing of analog signals
2/sc
)( jHaa
![Page 19: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/19.jpg)
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.
![Page 20: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/20.jpg)
![Page 21: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/21.jpg)
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 )
![Page 22: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/22.jpg)
:
)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
![Page 23: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/23.jpg)
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')
![Page 24: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/24.jpg)
Supplement: band-pass sampling theorem :
Nff
fM
ff
fN
NMfff
LH
H
LH
H
LHs
int
)/1)((2
![Page 25: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/25.jpg)
BfH 5
![Page 26: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/26.jpg)
4fs
BfB H 54
)/1(2)(
2/
2/2
2
NMBN
NMB
N
BfBNff
fNf
HHs
Hs
![Page 27: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/27.jpg)
4.1 summary
1.representation in time domain of sampling
) ( | ) ( ] [nT x t x n xc nT t c
![Page 28: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/28.jpg)
2.changes in frequency domain caused by sampling
k
cj
k
scs
TkjXT
eX
kjXT
jX
)/)2((1
)(
))((1
)(
![Page 29: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/29.jpg)
3. understand reconstruction in frequency domain
![Page 30: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/30.jpg)
4. understand reconstruction in time domain
![Page 31: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/31.jpg)
5. sampling theorem
NNs
NNs
![Page 32: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/32.jpg)
Requirements and difficulties :frequency spectrum chart of sampling and reconstructioncomprehension and application of sampling theorem
![Page 33: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/33.jpg)
4.2 discrete-time processing of continuous-time signals
Figure 4.11
![Page 34: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/34.jpg)
Figure 4.12
EXAMPLE
T
TeHjHTj
eff/||0
/||)()(
conditions : LTI ; no aliasing or aliasing occurred outside the pass band of filters
![Page 35: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/35.jpg)
Figure 4.13
EXAMPLE
aliasing occurred outside the pass band of digital
filters satisfies the equivalent relation of frequency response mentioned before.
![Page 36: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/36.jpg)
4.3 continuous-time processing of discrete-time signal
Figure 4.16
![Page 37: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/37.jpg)
||)/()( forTjHeH cj
Figure 4.12
c
![Page 38: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/38.jpg)
jj eeH )(
EXAMPLE Ideal delay system : noninteger delay
![Page 39: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/39.jpg)
4.4 digital processing of analog signals
Figure 4.41
quantization and coding
![Page 40: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/40.jpg)
Figure 4.46(b)
Sampling and holding
![Page 41: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/41.jpg)
Figure 4.48
uniform quantization and
coding
)(25.16
numbersbit :,2/2
dBBSNR
BX Bm
![Page 42: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/42.jpg)
Figure 4.51
quantization error of 3BIT
quantization error of 8BIT
![Page 43: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/43.jpg)
0 量化前的电平
量化后的电平
nonuniform quantization
![Page 44: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/44.jpg)
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
![Page 45: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/45.jpg)
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
![Page 46: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/46.jpg)
Figure 4.53 D/A 过程
)()( tRth N
reconstruction
![Page 47: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/47.jpg)
Figure 4.5
![Page 48: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/48.jpg)
record the digital sound
![Page 49: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/49.jpg)
Influence caused by sampling rate and quantizing bits
![Page 50: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/50.jpg)
Different tones require different sampling
rates.
![Page 51: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/51.jpg)
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
![Page 52: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/52.jpg)
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;
![Page 53: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/53.jpg)
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
![Page 54: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/54.jpg)
4.5.1 sampling rate reduction by an integer factor(downsampling, decimation)
:
][][
compressorratesamplinga
nMxnxd
Figure 4.20
![Page 55: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/55.jpg)
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
![Page 56: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/56.jpg)
)2/)2(2/
2
1)( jjj
d eXeXeX
)( jdeX
)( jeX
202
202
2/1
Figure 4.21(c)(d)
EXAMPLE M=2
![Page 57: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/57.jpg)
)3/)4()3/)2(3/
3
1)( jjjj
d eXeXeXeX
)(jeX
)(j
d eX 202
202
EXAMPLE M=3
![Page 58: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/58.jpg)
Figure 4.22(b)(c)
EXAMPLE M=3 , aliasing
frequency spectrum after decimation : period=2π ,M times wider , 1/M times higher
![Page 59: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/59.jpg)
Figure 4.23
MN / Condition to avoid aliasing :
Total downsampling system : Total system
![Page 60: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/60.jpg)
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
![Page 61: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/61.jpg)
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
![Page 62: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/62.jpg)
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
![Page 63: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/63.jpg)
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
![Page 64: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/64.jpg)
EXAMPLEtime-domain process of
mirror-image filter
![Page 65: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/65.jpg)
Figure 4.27
Use linear interpolation actually
![Page 66: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/66.jpg)
4.5.3 changing the sampling rate by a noninteger factor
Figure 4.28
M
Lff ss '
![Page 67: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/67.jpg)
4,3
300'400
ML
HztosignalsHzchangeEXAMPLE
![Page 68: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/68.jpg)
)( jeX
202)(),( jeiXjeeX
24/02
202
)( jeY
Advantages of decimation after interpolation :1.Combine antialiasing and reverse mirror-image filter2.Lossless information for upsampling
![Page 69: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/69.jpg)
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.
![Page 70: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/70.jpg)
FIGURE 4.44
![Page 71: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/71.jpg)
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.
![Page 72: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/72.jpg)
][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
![Page 73: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/73.jpg)
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.
![Page 74: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/74.jpg)
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
::
![Page 75: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/75.jpg)
decimation an d interpolation to realize pitch scale
4.pitch scale : decimation or interpolation , sampling rate of reconstruction is not changed.
![Page 76: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/76.jpg)
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
![Page 77: 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.](https://reader036.fdocument.pub/reader036/viewer/2022062308/56649c765503460f9492b555/html5/thumbnails/77.jpg)
exercises
4.15 (b)(c)4.24(a)(b)4.26 only for ωh= π /4