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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 Definition Classification of sequence Basic sequences Period of sequence Symmetry of sequence Energy of sequence The basic operations of sequences

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Definition EXAMPLE Enumerative representation Function representation

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

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10 1. Unit sample sequence 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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Period of sequence

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16 Three kinds of period of sequence

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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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Energy of sequence

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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 ); 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 =

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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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summary Definition Classification of sequence Basic sequences Period of sequence Symmetry of sequence Energy of sequence 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 Definition ： input-output description of systems Classification of discrete-time system Linear time-invariant system （ LTI ） Linear constant-coefficient difference equation Direct implementation of discrete-time system

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definition ： input-output description of systems the impulse response

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39 EXAMPLE

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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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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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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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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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summary Definition ： input-output description of systems Classification of discrete-time system Linear time-invariant system （ LTI ） Linear constant-coefficient difference equation 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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frequency-domain representation of discrete-time signal and system definition of fourier transform frequency response of system 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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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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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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properties of fourier transform

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

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