Post on 14-Dec-2015
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