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Digital Communication Vector Space concept

Digital Communication Vector Space concept. Signal space. Signal Space Inner Product Norm Orthogonality Equal Energy Signals Distance Orthonormal Basis Vector Representation Signal Space Summary. Signal Space. S(t). S=(s1,s2,…). Inner Product (Correlation) Norm (Energy)

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Digital Communication Vector Space concept

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  1. Digital Communication Vector Space concept 1

  2. Signal space • Signal Space • Inner Product • Norm • Orthogonality • Equal Energy Signals • Distance • Orthonormal Basis • Vector Representation • Signal Space Summary 2

  3. Signal Space S(t) S=(s1,s2,…) • Inner Product (Correlation) • Norm (Energy) • Orthogonality • Distance (Euclidean Distance) • Orthogonal Basis 3

  4. ONLY CONSIDER SIGNALS, s(t) T t Energy 4

  5. Inner Product - (x(t), y(t)) Similar to Vector Dot Product 5

  6. Example A T t -A 2A A/2 t T 6

  7. Norm - ||x(t)|| Similar to norm of vector A T -A 7

  8. Orthogonality A T -A Y(t) B Similar to orthogonal vectors T 8

  9. X(t) • ORTHONORMAL FUNCTIONS { T Y(t) T 9

  10. Correlation Coefficient 1    -1 =1 when x(t)=ky(t) (k>0) • In vector presentation 10

  11. Example Y(t) X(t) 10A A t t -A T T/2 7T/8 Now, shows the “real” correlation 11

  12. Distance, d • For equal energy signals • =-1 (antipodal) • =0 (orthogonal) • 3dB “better” then orthogonal signals 12

  13. Equal Energy Signals • To maximize d (antipodal signals) • PSK(phase Shift Keying) 13

  14. EQUAL ENERGY SIGNALS • ORTHOGONAL SIGNALS (=0) PSK (Orthogonal Phase Shift Keying) (Orthogonal if 14

  15. Signal Space summary • Inner Product • Norm ||x(t)|| • Orthogonality 15

  16. Corrolation Coefficient,  • Distance, d 16

  17. Modulation QAM BPSK QPSK BFSK 17

  18. Modulation • BPSK • QPSK • MPSK • QAM • Orthogonal FSK • Orthogonal MFSK • Noise • Probability of Error Modulation 18

  19. Binary Phase Shift Keying – (BPSK) - 19

  20. Binary antipodal signals vector presentation • Consider the two signals: The equivalent low pass waveforms are: 20

  21. The vector representation is – Signal constellation. 21

  22. The cross-correlation coefficient is: The Euclidean distance is: Two signals with cross-correlation coefficient of -1 are called antipodal 22

  23. Multiphase signals • Consider the M-ary PSK signals: The equivalent low pass waveforms are: 23

  24. The vector representation is: Or in complex-valued form as: 24

  25. Their complex-valued correlation coefficients are : and the real-valued cross-correlation coefficients are: The Euclidean distance between pairs of signals is: 25

  26. The minimum distance dmin corresponds to the case which | m-k |=1 26

  27. Quaternary PSK - QPSK (00) (10) (11) (01) * 27

  28. X(t) 28

  29. (00) (10) (11) (01) 29

  30. Exrecise 30

  31. MPSK 31

  32. 32

  33. Multi-amplitude Signal 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 . 33

  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 M=2 0 M=4 0 Signal-space diagram for M-ary PAM signals . 34

  35. The minimum distance between a pair signals 35

  36. Multi-Amplitude MultiPhase signalsQAM Signals A quadrature amplitude-modulated (QAM) signal or a quadrature-amplitude-shift-keying (QASK) is represented as Where and are the information bearing signal amplitudes of the quadrature carriers and u(t)= . 36

  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 37

  38. d QAM (Quadrature Amplitude Modulation) 38

  39. d QAM=QASK=AM-PM Exrecise 39

  40. M=256 M=128 M=64 M=32 M=16 M=4 + 40

  41. For an M - ary QAM Square Constellation In general for large M - adding one bit requires 6dB more energy to maintain same d . 41

  42. Binary orthogonal signals Consider the two signals Where either fc=1/T or fc>>1/T, so that Since Re(p12)=0, the two signals are orthogonal. 42

  43. The equivalent lowpass waveforms: The vector presentation: Which correspond to the signal space diagram Note that 43

  44. We observe that the vector representation for the equivalent lowpass signals is Where 44

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

  46. The real part of is 0 46

  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. 47

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

  49. Orthogonal FSK(Orthogonal Frequency Shift Keying) 51

  50. “0” “1” 52

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