1 of 71

Discrete Signals

1

Most slides from (edX) Discrete Time Signals and Systems by Prof. Richard G. Baraniuk

2 of 71

Discrete Time Signals

  •  

2

3 of 71

Plot Real Signals

  • Continuous signal

3

4 of 71

Plot Real Signals

  • Discrete signals

4

5 of 71

Discrete Signal Properties

5

6 of 71

Finite/Infinite Length Signals

  •  

6

7 of 71

Periodic Signals

  •  

7

8 of 71

Periodic Signals

  •  

8

9 of 71

Modular Arithmetic

  •  

9

10 of 71

Finite-Length and Periodic Signals

  • Finite-length and periodic signals are equivalent
    • All of the information in a periodic signal is contained in one period (of finite length)
    • Any finite-length signal can be periodized
    • Conclusion: We will think of finite-length signals and periodic signals interchangeably

10

11 of 71

Finite-Length and Periodic Signals

  •  

11

12 of 71

Finite-Length and Periodic Signals

  •  

12

13 of 71

Shifting Finite-Length Signals

  •  

13

14 of 71

Circular Time Reversal

  •  

14

15 of 71

Key Discrete Signals

15

16 of 71

Delta Function

  •  

16

17 of 71

Delta Function Sample

  •  

17

18 of 71

Unit Step Function

  •  

18

19 of 71

Unit Step Selects Part of a Signal

  •  

19

20 of 71

Real Exponential

  •  

20

21 of 71

Sinusoid Signals

  •  

21

22 of 71

Sinusoid Signals

  •  

22

23 of 71

Phase of Sinusoid

  •  

23

24 of 71

Complex Sinusoid

24

25 of 71

Complex Number

​

​

​

​

​

  • Adding

​

​

​

​

25

26 of 71

Euler’s Formula

26

27 of 71

Complex Number

​

​

​

​

​

  • Multiplying

​

27

28 of 71

Plot Complex Signals

  •  

28

phase

29 of 71

 

  •  

29

30 of 71

Sinusoidal Functions from Circular Motions

30

31 of 71

Sinusoidal Functions from Circular Motions

31

32 of 71

Discrete Sinusoids

  • Discrete Sinusoids

32

33 of 71

Visualize the Discrete Sinusoidals

33

34 of 71

Visualize the Discrete Sinusoidals

  •  

34

35 of 71

Visualize the Discrete Sinusoidals

  •  

35

36 of 71

Visualize the Discrete Sinusoidals

  •  

36

37 of 71

Visualize the Discrete Sinusoidals

  •  

37

38 of 71

Visualize the Discrete Sinusoidals

  •  

38

39 of 71

Visualize the Discrete Sinusoidals

  •  

39

40 of 71

Visualize the Discrete Sinusoidals

  •  

40

41 of 71

Visualize the Discrete Sinusoidals

  •  

41

42 of 71

Visualize the Discrete Sinusoidals

  •  

42

43 of 71

Aliasing

43

44 of 71

Aliasing of Discrete Sinusoids

  •  

44

45 of 71

Alias-free Frequencies in Discrete Sinusoids

  •  

45

46 of 71

Low and High Frequencies in Discrete Sinusoids

  •  

46

47 of 71

Low and High Frequencies in Discrete Sinusoids

  • Which one is a higher frequency?

47

48 of 71

Frequency in Discrete Sinusoids

48

49 of 71

Aliasing

49

50 of 71

Aliasing

50

51 of 71

Aliasing: Wheel

51

52 of 71

Visual Matrix of Discrete Sinusoids

52

53 of 71

Visual Matrix of Discrete Sinusoids

53

54 of 71

Complex Exponential Signals with Damping

54

55 of 71

Complex Exponential Signals

  •  

55

56 of 71

Damped Free Oscillation

56

PHY245: Damped Mass On A Spring, https://www.youtube.com/watch?v=ZqedDWEAUN4

57 of 71

Plot Complex Signals

  • Rectangular form

​

​

​

​

57

58 of 71

Plot Complex Signals

  • Polar form

​

​

​

​

​

58

59 of 71

Signals are Vectors

59

60 of 71

Signals are Vectors

  •  

60

61 of 71

Transpose of a Vector

  •  

61

62 of 71

Transpose in MATLAB

  • Be careful

62

63 of 71

Matrix Multiplication as Linear Combination

  •  

63

64 of 71

Matrix Multiplication as Linear Combination

  •  

64

65 of 71

Inner Product

  •  

65

66 of 71

Orthogonal Signals

  • Two sets of orthogonal signals

​

66

67 of 71

Harmonic Sinusoids are Orthogonal

  •  

67

68 of 71

Harmonic Sinusoids are Orthogonal

68

69 of 71

Harmonic Sinusoids are Orthogonal

69

70 of 71

Normalized Harmonic Sinusoids

70

71 of 71

Matrix Multiplication as a Sequence of Inner Products of Rows

  •  

71