Chapter 2 Discrete-time signals and systems

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

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

3. Enumerative representation Function representation 2.1.1 Definition EXAMPLE

4. 0.5 2 1 0 0 -1 -0.5 -2 -1 -3 0 5 10 -2 0 2 4 6 Graphical representation

5. Generate and plot the sequence in MATLAB 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,'.')

6. EXAMPLE Sampling the analog waveform Figure 2.2

7. Display the wav speech signal in ULTRAEDIT

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

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

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

11.  4.  Exponential sequence

13. 5. Sinusoidal sequence

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

15. 2.1.4 Period of sequence

16. Three kinds of period of sequence

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

19. EXAMPLE 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) Real sequences can be decomposed into two symmetrical sequences.

20. EXAMPLE Complex sequences can be decomposed into two symmetrical sequences. 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))

21. 2.1.6 Energy of sequence

22. 2.1.7 The basic operations of sequences

23. Basic operations of sequences

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

25. The matlab codes on the addition of two sequences EXAMPLE

26. 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<=max(n1)))=x1;y2((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

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

28. EXAMPLE 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)

29. 8.crosscorrelation: aotocorrelation:

31. example：correlation detection in digital audio watermark

34. 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

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

36. 2.2 Discrete-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

37. 2.2.1 definition：input-output description of systems the impulse response

38. EXAMPLE

39. 2.2.2 classification of discrete-time system 1．Memoryless (static) system the output depends only on the current input. 2．Linear system 3．Time-invariant system： 4．Causal system： the output does not depend on the latter input. 5．Stable system：

40. 2.2.3 linear time-invariant system（LTI） How to get h[n] from the input and output：

41. the impulse response in LTI EXAMPLE

42. h[n] Properties of LTI Figure 2.12

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

44. 2.2.4 linear constant-coefficient difference equation 1.relation with input-output description and convolution For IIR，the latter two are consistent. EXAMPLE input-output description convolution description infinite items，unrealizable difference equation description Finite items, realizable

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

46. 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

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

48. 2.2.5. Direct implementation of discrete-time system EXAMPLE

49. EXAMPLE

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