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

4.1periodic sampling 1.ideal sample T ： sample period fs=1/T:sample rate Ωs=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 time normalization t  t/T=n

Figure 4.3 frequency spectrum change of ideal sample aliasing frequency No aliasing aliasing

Period =2πin time domain ： w=2.1πand w=0.1πare the same trigonometric function property

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

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

Reconstruct frequency : EXAMPLE Sampling frequency:8Hz

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

ideal reconstruction in time domain

Figure 4.9 EXAMPLE

understand aliasing from time- domain interpolation

3.Nyquist sampling theorems

Nyquist sampling theorems:

Figure 4.41 Digital processing of analog signals

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 ）

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 ：

4.1 summary 1.representation in time domain of sampling

2.changes in frequency domain caused by sampling

3. understand reconstruction in frequency domain

4. understand reconstruction in time domain

5. sampling theorem

Requirements and difficulties ： frequency spectrum chart of sampling and reconstruction comprehension and application of sampling theorem

4.2 discrete-time processing of continuous-time signals Figure 4.11

Figure 4.12 EXAMPLE 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

Figure 4.12 c

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

Figure 4.51 quantization error of 3BIT quantization error of 8BIT

nonuniform quantization

x 码书 codeword c 0 codeword c 1 codeword c 2 codeword c 3 index0 d(x,c 0 )=5 d(x,c 1 )=11 d(x,c 2 )=8 d(x,c 3 )=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 Code book C ＝｛ y 0, y 1, y 2, y 3 ｝ y 0 y 1 y 2 y 3 codeword y 1 is the most adjacent to x ， so it is coded by the index “01”. d(x,y 0 )=25 d(x,y 1 )=5 d(x,y 2 )=25 d(x,y 3 )=46 vector quantization

Figure 4.53 D/A 过程 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 ， frequency spectrum chart; comprehension and application of sampling theorem; frequency response in discrete-time processing system of continuous-time signals;

4.5 changing the sampling rate using discrete-time processing sampling rate reduction by an integer factor (downsampling, decimation) increasing the sampling rate by an integer factor (upsampling, interpolation) changing the sampling rate by a noninteger fact application of multirate signal processing

4.5.1 sampling rate reduction by an integer factor (downsampling, decimation) Figure 4.20

time-domain of downsampling ： decrease the data ， reduce the sampling rate frequency-domain of downsampling ： take aliasing into consideration M=2 ， fs’=fs/M,T’=MT

Figure 4.21(c)(d) EXAMPLE M=2

)(  j eX )(  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 Condition to avoid aliasing ： Total downsampling system ： Total system

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

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

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: frequency domain of reverse mirror-image filter

total upsampling system ： total system Figure 4.24 time-domain explanation of reverse mirror-image filter :slowly-changed signal by interpolation

EXAMPLE time-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

EXAMPLE

Advantages of decimation after interpolation ： 1.Combine antialiasing and reverse mirror-image filter 2.Lossless information for upsampling

4.5.4 application of multirate signal processing Figure 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

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.

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 y i [n] in different sub-bands.

example ： compression for M=2

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 sampling rate reduction by an integer factor increasing the sampling rate by an integer factor changing the sampling rate by a noninteger fact application of multirate signal processing

exercises 4.15 (b)(c) 4.24(a)(b) 4.26 only for ω h = π /4