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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.

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Presentation on theme: "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."— Presentation transcript:

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2 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

3 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

4 3 2.1.1 Definition EXAMPLE Enumerative representation Function representation

5 4 -20246 -3 -2 0 1 2 0510 -0.5 0 0.5 Graphical representation

6 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

7 6 Figure 2.2 EXAMPLE Sampling the analog waveform

8 7 Display the wav speech signal in ULTRAEDIT

9 8 The whole waveform local Blowup Display the wav speech signal in COOLEDIT Display the wav speech signal in

10 9 2.1.2 Classification of sequence Right-side Left-side Two-side Finite-length Noncausal Causal

11 10 1. Unit sample sequence 2.1.3 Basic sequences 2 . The unit step sequence 3 . The rectangular sequence

12 11 4. Exponential sequence

13 12

14 13 5. Sinusoidal sequence

15 14 For convenience, sinusoidal signals are usually expressed by exponential sequences. The relationship between ω and Ω :

16 15 2.1.4 Period of sequence

17 16 Three kinds of period of sequence

18 17 2.1.5 Symmetry of sequence Conjugate-symmetric sequence Conjugate-antisymmetric sequence

19 18

20 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.

21 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.

22 21 2.1.6 Energy of sequence

23 22 2.1.7 The basic operations of sequences

24 23 Basic operations of sequences

25 24 Original music sequence Original speech sequences sequences after vector addition sequences after scalar multiplication sequences after vector multiplication echo

26 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

27 26 EXAMPLE The matlab codes on the addition of two sequences

28 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

29 28 7.convolution sum: steps : turnover, shift, vector multiplication, addition

30 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

31 30 8. crosscorrelation: aotocorrelation:

32 31

33 32 example : correlation detection in digital audio watermark

34 33

35 34

36 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

37 36 requirements : judge the period of sequence ; calculate convolution with graphical and analytical evaluation. key : convolution

38 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

39 38 2.2.1 definition : input-output description of systems the impulse response

40 39 EXAMPLE

41 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.

42 41 2.2.3 linear time-invariant system ( LTI ) How to get h[n] from the input and output :

43 42 EXAMPLE the impulse response in LTI

44 43 Figure 2.12 Properties of LTI h[n]

45 44 classification of linear time-invariant system IIR: h[n]’s length is infinite the latter input the former input FIR must be stable 。

46 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.

47 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

48 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

49 48 3.computation of difference equations with homogeneous and particular solution

50 49 2.2.5. Direct implementation of discrete-time system EXAMPLE

51 50 EXAMPLE

52 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

53 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

54 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

55 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

56 55 EXAMPLE The intuitionistic meaning of frequency- domain representation of signals

57 56 The intuitionistic meaning of frequency- domain representation of systems

58 57 EXAMPLE The effect of lowpass and highpass filters to image signals

59 58 Frequency-domain analysis of de-noise process through bandstop filter

60 59 Derivation of Fourier transform

61 60 2.3.1 definition of fourier transform arbitrary phase

62 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

63 62 EXAMPLE Fourier transforms of non- absolutely summable or non- square summable signals

64 63 2.3.2 frequency response of system

65 64 EXAMPLE Ideal filter in frequency and time domain

66 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

67 66 Eigenfunction and steady-state response : Steady-state response transient response

68 67 Figure 2.20 causal FIR system acts on causal signal Causal and stable IIR system acts on causal signal

69 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

70 69 2.3.3 properties of fourier transform

71 70

72 71

73 72

74 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

75 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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