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
Classification of sequence
Basic sequences
Period of sequence
2.1.5 Symmetry of sequence
Energy of sequence
2.1.7 The basic operations of sequences

3. 2.1.1 Definition EXAMPLE Enumerative representation
Function representation

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

5. Generate and plot the sequence in MATLAB
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. Sampling the analog waveform
Figure 2.2
EXAMPLE
Sampling the analog waveform

7. Display the wav speech signal in ULTRAEDIT

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

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. Period of sequence

16. Three kinds of period of sequence

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

19. Real sequences can be decomposed into two symmetrical sequences.
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. Complex sequences can be decomposed into two symmetrical sequences.
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. 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 processions
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.

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

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<=max(n1)))=x1;y2((m>=min(n2))&(m<=max(n2)))=x2;
y=3*y1+y2; stem(m,y)
Output:y =

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

29. 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)

30. 8.crosscorrelation:
aotocorrelation:

32. example：correlation detection in digital audio watermark

35. 2.1 summary
Definition
Classification of sequence
Basic sequences
Period of sequence
Symmetry of sequence
Energy of sequence
2.1.7 The basic operations of sequences

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

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

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

39. EXAMPLE

40. 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：

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

42. the impulse response in LTI
EXAMPLE

43. Properties of LTI
Figure 2.12
h[n]

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

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

46. 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.

47. EXAMPLE 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

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

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

50. EXAMPLE

51. The matlab codes on the direct realization of LTI
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])

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

53. judge the type of a system（from the relationship between
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 difference difference , recursion
stability stable maybe stable

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

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

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

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

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

59. Derivation of Fourier transform

60. 2.3.1 definition of fourier transform
arbitrary phase

61. Matlab codes to draw the frequency chart of signals
EXAMPLE
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')
Matlab codes to draw the frequency chart of signals

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

63. 2.3.2 frequency response of system

64. Ideal filter in frequency and time domain
EXAMPLE

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

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

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

68. example of steady-state response
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]

69. 2.3.3 properties of fourier transform

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

74. exercises： 2.35 2.45 2.57 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：