Title: Chapter 4 sampling of continous-time signals
1Chapter 4 sampling of continous-time signals
24.1 periodic sampling1.ideal sample
Tsample period fs1/Tsample rate Os2p/Tsample
rate
3Figure 4.1 ideal continous-time-to-discrete-time(C
/D)converter
4time normalization t?t/Tn
Figure 4.2(a) mathematic model for ideal C/D
5Figure 4.3
frequency spectrum change of ideal sample
No aliasing
aliasing
aliasing frequency
6Period 2pin time domain w2.1pand w0.1pare the
same
trigonometric function property
7high frequency is changed into low frequency in
time domainw1.1p and w0.9pare the same
trigonometric function property
82.ideal reconstruction
Figure 4.10(b) ideal D/C converter
9ideal reconstruction in frequency domain
Figure 4.4
10Take sinusoidal signal for example to understand
aliasing from frequency domain
Figure 4.5
EXAMPLE
11EXAMPLE
Sampling frequency8Hz
Reconstruct frequency
12Figure 4.10(a) mathematic model for ideal D/C
13ideal reconstruction in time domain
14EXAMPLE
Figure 4.9
15understand aliasing from time-domain interpolation
EXAMPLE
163.Nyquist sampling theorems
17Nyquist sampling theorems
18(No Transcript)
19examples 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.
20(No Transcript)
21examples 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
22Matlab codes to realize interpolation
EXAMPLE
23T0.1 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
')
24Supplement band-pass sampling theorem
25(No Transcript)
26(No Transcript)
274.1 summary
1.representation in time domain of sampling
28 2.changes in frequency domain caused by sampling
29 3. understand reconstruction in frequency domain
304. understand reconstruction in time domain
31 5. sampling theorem
32Requirements and difficulties frequency
spectrum chart of sampling and reconstruction com
prehension and application of sampling theorem
334.2 discrete-time processing of
continuous-time signals
34conditionsLTIno aliasing or aliasing occurred
outside the pass band of filters
EXAMPLE
35EXAMPLE
aliasing occurred outside the pass band of
digital filters satisfies the equivalent
relation of frequency response mentioned before.
Figure 4.13
364.3 continuous-time processing of discrete-time
signal
37(No Transcript)
38EXAMPLE
Ideal delay systemnoninteger delay
394.4 digital processing of analog signals
40Sampling and holding
Figure 4.46(b)
41Figure 4.48
42Figure 4.51
quantization error of 3BIT
quantization error of 8BIT
43nonuniform quantization
44vector quantization
45vector quantization
46reconstruction
Figure 4.53 D/A??
47Figure 4.5
48record the digital sound
49Influence caused by sampling rate and quantizing
bits
50Different tones require different sampling rates.
514.14.4 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
52Requirements 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
534.5 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
544.5.1 sampling rate reduction by an integer
factor (downsampling, decimation)
55time-domain of downsampling decrease the
data,reduce the sampling rate
M2,fsfs/M,TMT
56EXAMPLE
M2
57EXAMPLE
M3
58EXAMPLE
M3,aliasing
Figure 4.22(b)(c)
frequency spectrum after decimationperiod2p,M
times wider,1/M times higher
59Condition to avoid aliasing
Total downsampling systemTotal system
Figure 4.23
604.5.2 increasing the sampling rate by an integer
factor (upsampling, interpolation)
61time-domain of upsampling increase the
data,raise the sampling rate
L2,fsLfs,TT/L
62EXAMPLE
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
63time-domain explanation of reverse mirror-image
filter slowly-changed signal by interpolation
64EXAMPLE
time-domain process of mirror-image filter
65Use linear interpolation actually
664.5.3 changing the sampling rate by a noninteger
factor
67EXAMPLE
68Advantages of decimation after interpolation 1.Co
mbine antialiasing and reverse mirror-image
filter 2.Lossless information for upsampling
694.5.4 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
70FIGURE 4.44
712.reconstruction systemreplace high-powered
analog reconstructing filter with interpolation,
high-powered digital reverse mirror-image filter
and low-powered analog analog reconstructing
filter.
723. filter bank
analysis and synthesis of sub band
73In 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.
74examplecompression for M2
754.pitch scaledecimation or interpolation
,sampling rate of reconstruction is not changed.
decimation an d interpolation to realize pitch
scale
764.5 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
77exercises
4.15 (b)(c) 4.24(a)(b) 4.26 only for ?h p /4