# Chapter 4 sampling of continous-time signals 4.5 changing the sampling rate using discrete-time...

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Slide 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 of continuous-time signals 4.3 continuous-time processing of discrete-time signal 4.4 digital processing of analog signals Slide 3 4.1periodic sampling 1.ideal sample T sample period fs=1/T:sample rate s=2/T:sample rate Slide 4 Figure 4.1 ideal continous-time-to-discrete-time(C/D)converter Slide 5 Figure 4.2(a) mathematic model for ideal C/D time normalization t t/T=n Slide 6 Figure 4.3 frequency spectrum change of ideal sample aliasing frequency No aliasing aliasing Slide 7 Period =2in time domain w=2.1and w=0.1are the same trigonometric function property Slide 8 high frequency is changed into low frequency in time domain w=1.1 and w=0.9are the same trigonometric function property Slide 9 2.ideal reconstruction Figure 4.10(b) ideal D/C converter Slide 10 Figure 4.4 ideal reconstruction in frequency domain Slide 11 Figure 4.5 EXAMPLE Take sinusoidal signal for example to understand aliasing from frequency domain Slide 12 Reconstruct frequency : EXAMPLE Sampling frequency:8Hz Slide 13 Figure 4.10(a) mathematic model for ideal D/C Slide 14 ideal reconstruction in time domain Slide 15 Figure 4.9 EXAMPLE Slide 16 understand aliasing from time- domain interpolation Slide 17 3.Nyquist sampling theorems Slide 18 Nyquist sampling theorems: Slide 19 Figure 4.41 Digital processing of analog signals Slide 20 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. Slide 21 Slide 22 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 Slide 23 EXAMPLE Matlab codes to realize interpolation Slide 24 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') Slide 25 Supplement: band-pass sampling theorem Slide 26 Slide 27 Slide 28 4.1 summary 1.representation in time domain of sampling Slide 29 2.changes in frequency domain caused by sampling Slide 30 3. understand reconstruction in frequency domain Slide 31 4. understand reconstruction in time domain Slide 32 5. sampling theorem Slide 33 Requirements and difficulties frequency spectrum chart of sampling and reconstruction comprehension and application of sampling theorem Slide 34 4.2 discrete-time processing of continuous-time signals Figure 4.11 Slide 35 Figure 4.12 EXAMPLE conditions LTI no aliasing or aliasing occurred outside the pass band of filters Slide 36 Figure 4.13 EXAMPLE aliasing occurred outside the pass band of digital filters satisfies the equivalent relation of frequency response mentioned before. Slide 37 4.3 continuous-time processing of discrete-time signal Figure 4.16 Slide 38 Figure 4.12 c Slide 39 EXAMPLE Ideal delay system noninteger delay Slide 40 4.4 digital processing of analog signals Figure 4.41 quantization and coding Slide 41 Figure 4.46(b) Sampling and holding Slide 42 Figure 4.48 uniform quantization and coding Slide 43 Figure 4.51 quantization error of 3BIT quantization error of 8BIT Slide 44 nonuniform quantization Slide 45 x 4 33 2 3 1 3 2 2 2 1 3 4 3 4 1 1 1 3 4 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 x vector quantization Slide 46 Example: image coding Initial image block 4 gray-levels dimentions k=44=16 x 0 1 2 3 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 Slide 47 Figure 4.53 D/A reconstruction Slide 48 Figure 4.5 Slide 49 record the digital sound Slide 50 Influence caused by sampling rate and quantizing bits Slide 51 Different tones require different sampling rates. Slide 52 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 Slide 53 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; Slide 54 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 Slide 55 4.5.1 sampling rate reduction by an integer factor (downsampling, decimation) Figure 4.20 Slide 56 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 Slide 57 Figure 4.21(c)(d) EXAMPLE M=2 Slide 58 )( j eX )( j d eX 202 202 EXAMPLE M=3 Slide 59 Figure 4.22(b)(c) EXAMPLE M=3 aliasing frequency spectrum after decimation period=2 M times wider 1/M times higher Slide 60 Figure 4.23 Condition to avoid aliasing Total downsampling system Total system Slide 61 4.5.2 increasing the sampling rate by an integer factor (upsampling, interpolation) Slide 62 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 Slide 63 Figure 4.25 EXAMPLE L=2 transverse axis is 1/L timer shorter magnitude has no change. L mirror images in a period. Period=2 also period =2 /L Take T=T/L as D/C: frequency domain of reverse mirror-image filter Slide 64 total upsampling system total system Figure 4.24 time-domain explanation of reverse mirror-image filter :slowly-changed signal by interpolation Slide 65 EXAMPLE time-domain process of mirror-image filter Slide 66 Figure 4.27 Use linear interpolation actually Slide 67 4.5.3 changing the sampling rate by a noninteger factor Figure 4.28 Slide 68 EXAMPLE Slide 69 Advantages of decimation after interpolation 1.Combine antialiasing and reverse mirror-image filter 2.Lossless information for upsampling Slide 70 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. Slide 71 FIGURE 4.44 Slide 72 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. Slide 73 analysis and synthesis of sub band 3. filter bank Slide 74 In MP3, M=32 sub-band analysis filter bank is 32 equi-band filters with center frequency uniformly distributed from 0 to MP3 coders use different quantization to realize compression for signals y i [n] in different sub-bands. Slide 75 example compression for M=2 Slide 76 decimation an d interpolation to realize pitch scale 4.pitch scale decimation or interpolation sampling rate of reconstruction is not changed. Slide 77 4.5 summary requirement frequency spectrum chart of interpolation and decimation 4.5.1 sampling rate reduction by an integer factor 4.5.2 increasing the sampling rate by an integer factor 4.5.3 changing the sampling rate by a noninteger fact 4.5.4 application of multirate signal processing Slide 78 exercises 4.15 (b)(c) 4.24(a)(b) 4.26 only for h = /4