Chapter 4 sampling of continous-time signals

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Chapter 4 sampling of continous-time signals
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4.1 periodic sampling1.ideal sample
Tsample period fs1/Tsample rate Os2p/Tsample
rate
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Figure 4.1 ideal continous-time-to-discrete-time(C
/D)converter
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time normalization t?t/Tn
Figure 4.2(a) mathematic model for ideal C/D
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Figure 4.3
frequency spectrum change of ideal sample
No aliasing
aliasing
aliasing frequency
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Period 2pin time domain w2.1pand w0.1pare the
same
trigonometric function property
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high frequency is changed into low frequency in
time domainw1.1p and w0.9pare the same
trigonometric function property
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2.ideal reconstruction
Figure 4.10(b) ideal D/C converter
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ideal reconstruction in frequency domain
Figure 4.4
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Take sinusoidal signal for example to understand
aliasing from frequency domain
Figure 4.5
EXAMPLE
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EXAMPLE
Sampling frequency8Hz
Reconstruct frequency
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Figure 4.10(a) mathematic model for ideal D/C
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ideal reconstruction in time domain
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EXAMPLE
Figure 4.9
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understand aliasing from time-domain interpolation
EXAMPLE
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3.Nyquist sampling theorems
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Nyquist sampling theorems
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examples of sampling theorem(1)
The highest frequency of analog signal ,which wav
file with sampling rate 16kHz can show , is
8kHz
The higher sampling rate of audio files, the
better fidelity.
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examples of sampling theorem(2)
according to what you know about the
sampling rate of MP3 file,judge the sound we can
feel frequency range( )
(A)2044.1kHz (B)2020kHz
(C)204kHz (D)208kHz
B
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Matlab codes to realize interpolation
EXAMPLE
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T0.1 n010 xcos(10pinT) stem(n,x) dt
0.001 tones(11,1) 0dt1 nn'ones(1,1/dt
1) yxsinc((t-nT)/T) hold on plot(t/T,y,'r
')
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Supplement band-pass sampling theorem
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4.1 summary
1.representation in time domain of sampling
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2.changes in frequency domain caused by sampling
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3. understand reconstruction in frequency domain
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4. understand reconstruction in time domain
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5. sampling theorem
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Requirements and difficulties frequency
spectrum chart of sampling and reconstruction com
prehension and application of sampling theorem
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4.2 discrete-time processing of
continuous-time signals
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conditionsLTIno aliasing or aliasing occurred
outside the pass band of filters
EXAMPLE
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EXAMPLE
aliasing occurred outside the pass band of
digital filters satisfies the equivalent
relation of frequency response mentioned before.
Figure 4.13
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4.3 continuous-time processing of discrete-time
signal
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EXAMPLE
Ideal delay systemnoninteger delay
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4.4 digital processing of analog signals
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Sampling and holding
Figure 4.46(b)
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Figure 4.48
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Figure 4.51
quantization error of 3BIT
quantization error of 8BIT
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nonuniform quantization
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vector quantization
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vector quantization
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reconstruction
Figure 4.53 D/A??
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Figure 4.5
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record the digital sound
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Influence caused by sampling rate and quantizing
bits
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Different tones require different sampling rates.
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4.14.4 summary

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

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Requirements and difficulties sampling
processing in time and frequency domain,frequency
spectrum chart comprehension and application of
sampling theorem frequency response in
discrete-time processing system of
continuous-time signals
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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
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4.5.1 sampling rate reduction by an integer
factor (downsampling, decimation)
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time-domain of downsampling decrease the
data,reduce the sampling rate
M2,fsfs/M,TMT
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EXAMPLE
M2
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EXAMPLE
M3
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EXAMPLE
M3,aliasing
Figure 4.22(b)(c)
frequency spectrum after decimationperiod2p,M
times wider,1/M times higher
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Condition to avoid aliasing
Total downsampling systemTotal system
Figure 4.23
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4.5.2 increasing the sampling rate by an integer
factor (upsampling, interpolation)
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time-domain of upsampling increase the
data,raise the sampling rate
L2,fsLfs,TT/L
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EXAMPLE
Figure 4.25
L2
frequency domain of reverse mirror-image filter
transverse axis is 1/L timer shorter,magnitude
has no change. L mirror images in a period.
Period2p,also period 2 p /L
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time-domain explanation of reverse mirror-image
filter slowly-changed signal by interpolation
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EXAMPLE
time-domain process of mirror-image filter
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Use linear interpolation actually
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4.5.3 changing the sampling rate by a noninteger
factor
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EXAMPLE
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Advantages of decimation after interpolation 1.Co
mbine antialiasing and reverse mirror-image
filter 2.Lossless information for upsampling
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4.5.4 application of multirate signal processing
1.Sampling systemreplace 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 4.43
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FIGURE 4.44
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2.reconstruction systemreplace high-powered
analog reconstructing filter with interpolation,
high-powered digital reverse mirror-image filter
and low-powered analog analog reconstructing
filter.
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3. filter bank
analysis and synthesis of sub band
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In MP3, M32,sub-band analysis filter bank is 32
equi-band filters with center frequency uniformly
distributed from 0 to p
MP3 coders use different quantization to realize
compression for signals yin in different
sub-bands.
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examplecompression for M2
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4.pitch scaledecimation or interpolation
,sampling rate of reconstruction is not changed.
decimation an d interpolation to realize pitch
scale
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4.5 summary
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
requirement frequency spectrum chart of
interpolation and decimation
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exercises
4.15 (b)(c) 4.24(a)(b) 4.26 only for ?h p /4