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1 2.3 Frequency-domain representation of discrete-time signal and system Chapter 2 Discrete-time signals and systems 2.1 Discrete-time signals:sequences 2.2Discrete-time system

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2 2.1 Discrete-time signals:sequences 2.1.1 Definition 2.1.2 Classification of sequence 2.1.3 Basic sequences 2.1.4 Period of sequence 2.1.5 Symmetry of sequence 2.1.6 Energy of sequence 2.1.7 The basic operations of sequences

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3 2.1.1 Definition EXAMPLE Enumerative representation Function representation

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4 -20246 -3 -2 0 1 2 0510 -0.5 0 0.5 Graphical representation

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5 n=-1:5 x=[1,2,1.2,0,-1,-2,-2.5] stem(n,x, '.') n=0:9 y=0.9.^n.*cos(0.2*pi*n+pi/2) stem(n,y,'.') Generate and plot the sequence in MATLAB

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6 Figure 2.2 EXAMPLE Sampling the analog waveform

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7 Display the wav speech signal in ULTRAEDIT

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8 The whole waveform local Blowup Display the wav speech signal in COOLEDIT Display the wav speech signal in

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9 2.1.2 Classification of sequence Right-side Left-side Two-side Finite-length Noncausal Causal

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10 1. Unit sample sequence 2.1.3 Basic sequences 2 ． The unit step sequence 3 ． The rectangular sequence

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11 4. Exponential sequence

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13 5. Sinusoidal sequence

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14 For convenience, sinusoidal signals are usually expressed by exponential sequences. The relationship between ω and Ω ：

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15 2.1.4 Period of sequence

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16 Three kinds of period of sequence

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17 2.1.5 Symmetry of sequence Conjugate-symmetric sequence Conjugate-antisymmetric sequence

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19 n=[-5:5]; x=[0,0,0,0,0,1,2,3,4,5,6]; xe=(x+fliplr(x))/2; xo=(x-fliplr(x))/2; subplot(3,1,1) stem(n,x) subplot(3,1,2) stem(n,xe) subplot(3,1,3) stem(n,xo) EXAMPLE Real sequences can be decomposed into two symmetrical sequences.

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20 n=[-5:5]; x=zeros(1,11); x((n>=0)&(n<=5))=(1+j).^[0:5] xe=(x+conj(fliplr(x)))/2; xo=(x-conj(fliplr(x)))/2 subplot(3,2,1); stem(n,real(x)) subplot(3,2,2); stem(n,imag(x)) subplot(3,2,3); stem(n,real(xe)) subplot(3,2,4); stem(n,imag(xe)) subplot(3,2,5); stem(n,real(xo)) subplot(3,2,6); stem(n,imag(xo)) EXAMPLE Complex sequences can be decomposed into two symmetrical sequences.

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21 2.1.6 Energy of sequence

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22 2.1.7 The basic operations of sequences

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23 Basic operations of sequences

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24 Original music sequence Original speech sequences sequences after vector addition sequences after scalar multiplication sequences after vector multiplication echo

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25 x=wavread('test1.wav',36000); y=wavread('test2.wav ',36000); z=(x+y)/2.0; wavwrite(z,22050,'test3.wav') y1=y*0.5; wavwrite(y1,22050,'test4.wav') y2=zeros(36000,1); for i=2000:36000 y2(i)=y(i-2000+1); end y3=0.6*y+0.4*y2; wavwrite(y3,22050,'test5.wav') w=[0:1/36000:1-1/36000]'; y4=y.*w; wavwrite(y4,22050,'test6.wav') Vector addition realizes composition. scalar multiplication changes the volume. Delay, scalar multiplication and vector addition produce echo. vector multiplication realizes fade-in. The matlab codes on the processions

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26 EXAMPLE The matlab codes on the addition of two sequences

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27 n=[-4:2] ;x=[1,-2,4,6,-5,8,10] ; %x1[n]=x[n+2] n1=n-2;x1=x; %x2[n]=x[n-4] n2=n+4;x2=x; %y[n] m=[min(min(n1),min(n2)): max(max(n1),max(n2))] ; y1=zeros(1,length(m)) ;y2=y1; y1((m>=min(n1))&(m =min(n2))&(m<=max(n2)))=x2; y=3*y1+y2;stem(m,y) Output:y =3-6 12 18 -15 24 31 -2 4 6 -5 8 10

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28 7.convolution sum: steps ： turnover, shift, vector multiplication, addition

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29 nx=0:10;x=0.5.^nx; nh=-1:4;h=ones(1,length(nh)) y=conv(x,h);stem([min(nx)+min(nh):max(nx)+max(nh)],y) EXAMPLE

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30 8. crosscorrelation: aotocorrelation:

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32 example ： correlation detection in digital audio watermark

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35 2.1 summary 2.1.1 Definition 2.1.2 Classification of sequence 2.1.3 Basic sequences 2.1.4 Period of sequence 2.1.5 Symmetry of sequence 2.1.6 Energy of sequence 2.1.7 The basic operations of sequences

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36 requirements ： judge the period of sequence ; calculate convolution with graphical and analytical evaluation. key ： convolution

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37 2.2Discrete-time system 2.2.1 Definition ： input-output description of systems 2.2.2 Classification of discrete-time system 2.2.3 Linear time-invariant system （ LTI ） 2.2.4 Linear constant-coefficient difference equation 2.2.5. Direct implementation of discrete-time system

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38 2.2.1 definition ： input-output description of systems the impulse response

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39 EXAMPLE

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40 2.2.2 classification of discrete-time system 1 ． Memoryless (static) system the output depends only on the current input. 5 ． Stable system ： 2 ． Linear system 3 ． Time-invariant system ： 4 ． Causal system ： the output does not depend on the latter input.

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41 2.2.3 linear time-invariant system （ LTI ） How to get h[n] from the input and output ：

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42 EXAMPLE the impulse response in LTI

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43 Figure 2.12 Properties of LTI h[n]

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44 classification of linear time-invariant system IIR: h[n]’s length is infinite the latter input the former input FIR must be stable 。

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45 2.2.4 linear constant-coefficient difference equation 1.relation with input-output description and convolution EXAMPLE input-output description convolution description infinite items ， unrealizable difference equation description Finite items, realizable For IIR ， the latter two are consistent.

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46 EXAMPLE For FIR ， the followings are consistent For FIR and IIR ， difference equations are not exclusive. input-output description and difference equation description （ non-recursion ） Convolution description Another difference equation description ， recursion ， lower rank

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47 2.Recursive computation of difference equations ： For IIR, there needs N initial conditions, then,the solution is unique. For FIR, there needs no initial conditions. With initial-rest conditions (linear, time invariant, and causal), the solution is unique. EXAMPLE

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48 3.computation of difference equations with homogeneous and particular solution

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49 2.2.5. Direct implementation of discrete-time system EXAMPLE

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50 EXAMPLE

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51 B=1;A=[1,-1] n=[0:100];x=[n>=0]; y=filter(B,A,x);stem(n,y);axis([0,20,0,20]) The matlab codes on the direct realization of LTI EXAMPLE

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52 2.2 summary 2.2.1 Definition ： input-output description of systems 2.2.2 Classification of discrete-time system 2.2.3 Linear time-invariant system （ LTI ） 2.2.4 Linear constant-coefficient difference equation 2.2.5. Direct implementation of discrete-time system

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53 keys ： judge the type of a system （ from the relationship between the input and output, and from h[n] for LTI). the physics meaning of convolution representation for LTI ： the output signals are the weighted combination of the input signals ， h[n] is the weight 。 the similarities and differences between linear constant-coefficient difference equations and convolution representation ， recursive computation 。 the difference between IIR and FIR ： FIR IIR h[n]finite length infinite length y[n] 是 x[n] 的加权 finite items infinite items realization convolution or differencedifference, recursion stabilitystable maybe stable

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54 2.3 frequency-domain representation of discrete-time signal and system 2.3.1 definition of fourier transform 2.3.2 frequency response of system 2.3.3 properties of fourier transform

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55 EXAMPLE The intuitionistic meaning of frequency- domain representation of signals

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56 The intuitionistic meaning of frequency- domain representation of systems

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57 EXAMPLE The effect of lowpass and highpass filters to image signals

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58 Frequency-domain analysis of de-noise process through bandstop filter

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59 Derivation of Fourier transform

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60 2.3.1 definition of fourier transform arbitrary phase

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61 subplot(2,2,1);fplot('real(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);title(' 实部 ') subplot(2,2,2);fplot('imag(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);title(' 虚部 ') subplot(2,2,3);fplot('abs(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);title(' 幅度 ') subplot(2,2,4);fplot('angle(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]); title(' 相位 ') EXAMPLE Matlab codes to draw the frequency chart of signals

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62 EXAMPLE Fourier transforms of non- absolutely summable or non- square summable signals

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63 2.3.2 frequency response of system

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64 EXAMPLE Ideal filter in frequency and time domain

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65 h=[1,0,0,0,0,0,0,0,0,0.5] freqz(h,1) EXAMPLE Matlab codes to draw the frequency response of a system

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66 Eigenfunction and steady-state response ： Steady-state response transient response

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67 Figure 2.20 causal FIR system acts on causal signal Causal and stable IIR system acts on causal signal

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68 Sin(0.1*pi*n) h[n]=[1,1,1,1,1,1,1,1,1,1]/4 B=[1,0,1,0,1];A=[1,0.81,0.81,0.81] example of steady-state response

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69 2.3.3 properties of fourier transform

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73 2.3 summary requirements ： calculation of fourier transforms steady-state response linearity time shifting frequency shifting the convolution theorem windowing theorem Parseval’s theorem symmetry properties 2.3.1 definition of fourier transform 2.3.2 frequency response of system 2.3.3 properties of fourier transform

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74 Keys and difficulties ： the convolution theorem; the frequency spectrum of a real sequence is conjugate symmetric; the frequency spectrum of a conjugate symmetric sequence is a real function. exercises ： 2.35 2.45 2.57

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