2. Digital communication - vector approach Dr. Uri Mahlab 1 Digital Communication Vector Space concept

3. Digital communication - vector approach Dr. Uri Mahlab 2 Signal space n Signal Space Signal Space n Inner Product Inner Product n Norm Norm n Orthogonality Orthogonality n Equal Energy Signals Equal Energy Signals n Distance Distance n Orthonormal Basis Orthonormal Basis n Vector Representation Vector Representation n Signal Space Summary Signal Space Summary

4. Digital communication - vector approach Dr. Uri Mahlab 3 Signal Space S(t)S=(s1,s2,…) Inner Product (Correlation) Norm (Energy) Orthogonality Distance (Euclidean Distance) Orthogonal Basis

5. Digital communication - vector approach Dr. Uri Mahlab 4 Energy ONLY CONSIDER SIGNALS, s(t) T t

6. Digital communication - vector approach Dr. Uri Mahlab 5 Similar to Vector Dot Product Inner Product - (x(t), y(t))

7. Digital communication - vector approach Dr. Uri Mahlab 6 A -A-A 2A2A A/2 T T t t Example

8. Digital communication - vector approach Dr. Uri Mahlab 7 Norm - ||x(t)|| Similar to norm of vector T A -A-A

9. Digital communication - vector approach Dr. Uri Mahlab 8 Orthogonality Similar to orthogonal vectors T A -A-A T Y(t) B

10. Digital communication - vector approach Dr. Uri Mahlab 9 ORTHONORMAL FUNCTIONS { T T X(t) Y(t)

11. Digital communication - vector approach Dr. Uri Mahlab 10 Correlation Coefficient In vector presentation 1    -1  =  1 when x(t)=  ky(t) (k>0)

12. Digital communication - vector approach Dr. Uri Mahlab 11 Example Now,  shows the “real” correlation tt A -A-A T/27T/8 T 10A X(t) Y(t)

13. Digital communication - vector approach Dr. Uri Mahlab 12 Distance, d For equal energy signals  =-1 (antipodal) 3dB “better” then orthogonal signals  =0 (orthogonal)

14. Digital communication - vector approach Dr. Uri Mahlab 13 Equal Energy Signals PSK (phase Shift Keying) To maximize d (antipodal signals)

15. Digital communication - vector approach Dr. Uri Mahlab 14 EQUAL ENERGY SIGNALS ORTHOGONAL SIGNALS (  =0) PSK (Orthogonal Phase Shift Keying) (Orthogonal if

16. Digital communication - vector approach Dr. Uri Mahlab 15 Signal Space summary Inner Product Norm ||x(t)|| Orthogonality

17. Digital communication - vector approach Dr. Uri Mahlab 16 Corrolation Coefficient,  Distance, d

18. Digital communication - vector approach Dr. Uri Mahlab 17 Modulation

19. Digital communication - vector approach Dr. Uri Mahlab 18 Modulation  ModulationModulation  BPSKBPSK  QPSKQPSK  MPSKMPSK  QAMQAM  Orthogonal FSKOrthogonal FSK  Orthogonal MFSKOrthogonal MFSK  NoiseNoise  Probability of ErrorProbability of Error

20. Digital communication - vector approach Dr. Uri Mahlab 19 - Binary Phase Shift Keying – (BPSK)

21. Digital communication - vector approach Dr. Uri Mahlab 20 Binary antipodal signals vector presentation Consider the two signals: The equivalent low pass waveforms are:

22. Digital communication - vector approach Dr. Uri Mahlab 21 The vector representation is – Signal constellation.

23. Digital communication - vector approach Dr. Uri Mahlab 22 The cross-correlation coefficient is: The Euclidean distance is: Two signals with cross-correlation coefficient of -1 are called antipodal

24. Digital communication - vector approach Dr. Uri Mahlab 23 Multiphase signals Consider the M-ary PSK signals: The equivalent low pass waveforms are:

25. Digital communication - vector approach Dr. Uri Mahlab 24 The vector representation is: Or in complex-valued form as:

26. Digital communication - vector approach Dr. Uri Mahlab 25 Their complex-valued correlation coefficients are : and the real-valued cross-correlation coefficients are: The Euclidean distance between pairs of signals is:

27. Digital communication - vector approach Dr. Uri Mahlab 26 The minimum distance d min corresponds to the case which | m-k |=1

28. Digital communication - vector approach Dr. Uri Mahlab 27 (00) (01)(11) (10) * Quaternary PSK - QPSK

29. Digital communication - vector approach Dr. Uri Mahlab 28 X(t)

30. Digital communication - vector approach Dr. Uri Mahlab 29 (00) (01)(11) (10)

31. Digital communication - vector approach Dr. Uri Mahlab 30

32. Digital communication - vector approach Dr. Uri Mahlab 31 MPSK

33. Digital communication - vector approach Dr. Uri Mahlab 32

34. Digital communication - vector approach Dr. Uri Mahlab 33 Consider the M-ary PAM signals m=1,2,….,M Where this signal amplitude takes the discrete values (levels) m=1,2,….,M The signal pulse u(t), as defined is rectangular U(t)= But other pulse shapes may be used to obtain a narrower signal spectrum. Multi-amplitude Signal

35. Digital communication - vector approach Dr. Uri Mahlab 34 Clearly, this signals are one dimensional (N=1) and, hence, are represented by the scalar components M=1,2,….,M The distance between any pair of signal is 0 M=2 M=4 0 Signal-space diagram for M-ary PAM signals.

36. Digital communication - vector approach Dr. Uri Mahlab 35 The minimum distance between a pair signals

37. Digital communication - vector approach Dr. Uri Mahlab 36 Where and are the information bearing signal amplitudes of the quadrature carriers and u(t)=. A quadrature amplitude-modulated (QAM) signal or a quadrature-amplitude-shift-keying (QASK) is represented as Multi-Amplitude MultiPhase signals QAM Signals

38. Digital communication - vector approach Dr. Uri Mahlab 37 QAM signals are two dimensional signals and, hence, they are represented by the vectors The distance between a pair of signal vectors is k,m=1,2,…,M When the signal amplitudes take the discrete values In this case the minimum distance is

39. Digital communication - vector approach Dr. Uri Mahlab 38 QAM (Quadrature Amplitude Modulation) d

40. Digital communication - vector approach Dr. Uri Mahlab 39 QAM=QASK=AM-PM d

41. Digital communication - vector approach Dr. Uri Mahlab 40 M=256 M=128 M=64 M=32 M=16 M=4 +

42. Digital communication - vector approach Dr. Uri Mahlab 41 For an M - ary QAM Square Constellation In general for large M - adding one bit requires 6dB more energy to maintain same d.

43. Digital communication - vector approach Dr. Uri Mahlab 42 Binary orthogonal signals Consider the two signals Where either fc=1/T or fc>>1/T, so that Since Re(p 12 )=0, the two signals are orthogonal.

44. Digital communication - vector approach Dr. Uri Mahlab 43 The equivalent lowpass waveforms: The vector presentation: Which correspond to the signal space diagram Note that

45. Digital communication - vector approach Dr. Uri Mahlab 44 We observe that the vector representation for the equivalent lowpass signals is Where

46. Digital communication - vector approach Dr. Uri Mahlab 45 m=1,2,….,M This waveform are characterized as having equal energy and cross-correlation coefficients Let us consider the set of M FSK signals M-ary Orthogonal Signal

47. Digital communication - vector approach Dr. Uri Mahlab 46 0 The real part of is

48. Digital communication - vector approach Dr. Uri Mahlab 47 First, we observe that =0 when and. Since |m-k|=1 corresponds to adjacent frequency slots, represent the minimum frequency separation between adjacent signals for orthogonality of the M signals.

49. Digital communication - vector approach Dr. Uri Mahlab 48 Orthogonal signals for M=N=3 signal space diagram For the case in which,the FSK signals are equivalent to the N- dimensional vectors =(,0,0,…,0) =(0,,0,…,0) =(0,0,…,0, ) Where N=M. The distance between pairs of signals is all m,k Which is also the minimum distance.

50. Digital communication - vector approach Dr. Uri Mahlab 49 A set of M bi-orthogonal signals can be constructed from M/2 orthogonal signals by simply including the negatives of the orthogonal signals. Thus, we require N=M/2 dimensions for the construction of M bi-ortogonal signals. M=4M=6 Biorthogonal Signal

51. Digital communication - vector approach Dr. Uri Mahlab 50 We note that the correlation between any pair of waveforms is either or 0. The corresponding distances are or, with the latter being the minimum distance.

52. Digital communication - vector approach Dr. Uri Mahlab 51 Orthogonal FSK (Orthogonal Frequency Shift Keying)

53. Digital communication - vector approach Dr. Uri Mahlab 52 “0” “1”

54. Digital communication - vector approach Dr. Uri Mahlab 53 ORTHOGONAL MFSK

55. Digital communication - vector approach Dr. Uri Mahlab 54 All signals are orthogonal to each other

56. Digital communication - vector approach Dr. Uri Mahlab 55 How to generate signals

57. Digital communication - vector approach Dr. Uri Mahlab 56 0 T 2T 3T 4T 5T 6T +

58. Digital communication - vector approach Dr. Uri Mahlab 57 0 T 2T 3T 4T 5T 6T +

59. Digital communication - vector approach Dr. Uri Mahlab 58 0 T 2T 3T 4T 5T 6T +

60. Digital communication - vector approach Dr. Uri Mahlab 59 + IQ Modulator

61. Digital communication - vector approach Dr. Uri Mahlab 60 + IQ Modulator Pulse shaping filter

62. Digital communication - vector approach Dr. Uri Mahlab 61 NOISE

63. Digital communication - vector approach Dr. Uri Mahlab 62 What about Noise White Gaussian Noise TT The coefficients are random variables !

64. Digital communication - vector approach Dr. Uri Mahlab 63 WHITE GAUSSIAN NOISE (WGN) We write All are gaussian variables All are independent

65. Digital communication - vector approach Dr. Uri Mahlab 64 All have same probability distribution

66. Digital communication - vector approach Dr. Uri Mahlab 65 White Gaussian Noise has energy in every dimension

67. Digital communication - vector approach Dr. Uri Mahlab 66 Probability of Error for Binary Signaling The two signal waveforms are given as These waveforms are assumed to have equal energy E and their equivalent lowpass u m (t), m=1,2 are characterized by the complex-valued correlation coefficient ρ 12.

68. Digital communication - vector approach Dr. Uri Mahlab 67 The optimum demodulator forms the decision variables Or,equivalently And decides in favor of the signal corresponding to the larger decision variable.

69. Digital communication - vector approach Dr. Uri Mahlab 68 Lets see that the two expressions yields the same probability of error. Suppose the signal s 1 (t) is transmitted in the interval 0  t  T. The equivalent low-pass received signal is Substituting it into Um expression obtain Where Nm, m=1,2, represent the noise components in the decision variables,given by

70. Digital communication - vector approach Dr. Uri Mahlab 69 And. The probability of error is just the probability that the decision variable U2 exceeds the decision variable u1. But Lets define variable V as N 1r and N 2r are gaussian, so N 1 r-N 2r is also gaussian-distributed and, hence, V is gaussian- distributed with mean value

71. Digital communication - vector approach Dr. Uri Mahlab 70 And variance Where N0 is the power spectral density of z(t). The probability of error is now

72. Digital communication - vector approach Dr. Uri Mahlab 71 Where erfc(x) is the complementary error function, defined as It can be easily shown that

73. Digital communication - vector approach Dr. Uri Mahlab 72 Distance, d For equal energy signals  =-1 (antipodal) 3dB “better” then orthogonal signals  =0 (orthogonal)

74. Digital communication - vector approach Dr. Uri Mahlab 73 It is interesting to note that the probability of error P2 is expressed as Where d12 is the distance of the two signals. Hence,we observe that an increase in the distance between the two signals reduces the probability of error.

75. Digital communication - vector approach Dr. Uri Mahlab 74

76. Digital communication - vector approach Dr. Uri Mahlab 75