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beamforming.pptx

  1. Beamforming
  2. Tx1
  3. Tx1 cos(2šœ‹š‘“š‘”)
  4. Tx1 Tx2 š€ šŸ cos(2šœ‹š‘“š‘”)
  5. Tx1 Tx2 š€ šŸ cos(2šœ‹š‘“š‘”) Rx šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’• + š…
  6. Destructive superimposition Tx1 Tx2 š€ šŸ Zero signal cos 2šœ‹š‘“š‘” + cos 2šœ‹š‘“š‘” + šœ‹ = 0
  7. Tx1 Tx2 š€ šŸ Rx šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’•
  8. Constructive superimposition Tx1 Tx2 š€ šŸ Amplified signal (twice amplitude) cos 2šœ‹š‘“š‘” + cos 2šœ‹š‘“š‘” + 0 = 2cos(2šœ‹š‘“š‘”)
  9. Receiver at arbitrary location Tx1 Tx2 š€ šŸ šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’• + š“ Rx
  10. Arbitrary location, whatā€™s the path difference Path difference = ??
  11. Path difference Tx1 Tx2 š’… šœƒ šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’• + š“ Rx
  12. Path difference and phase difference Tx1 Tx2 š’… Path difference = šœƒ š’…š’„š’š’”(šœ½) šœ™(š‘ā„Žš‘Žš‘ š‘’ š‘‘š‘–š‘“š‘“š‘’š‘Ÿš‘’š‘›š‘š‘’) = 2šœ‹ šœ† āˆ— (š‘š‘Žš‘”ā„Ž š‘‘š‘–š‘“š‘“š‘’š‘Ÿš‘’š‘›š‘š‘’) šœ™ = 2šœ‹ šœ† āˆ— š’…š’„š’š’”(šœ½) šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’• + š“ Rx š‘š± šœ½ = šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’• + šŸš… š€ š’…š’„š’š’”(šœ½)
  13. (š‘‘ = šœ† 2 ) Radiation pattern: Rx amplitude as a function of angle
  14. (š‘‘ = šœ†) Radiation pattern: Rx amplitude as a function of angle
  15. Radiation pattern: Rx amplitude as a function of angle (š‘‘ = 2šœ†)
  16. (š‘‘ = šœ† 2 ) Radiation pattern: Rx amplitude as a function of angle šœšØš¬ šŸš…š’‡š’• + šœšØš¬ šŸš…š’‡š’• + š“
  17. (š‘‘ = šœ† 2 ) Radiation pattern: Rx amplitude as a function of angle šœšØš¬ šŸš…š’‡š’• + š“š’Šš’ + šœšØš¬ šŸš…š’‡š’• + š“ The initial phases can be controlled
  18. Radiation pattern: Rx amplitude as a function of angle (š‘‘ = šœ† 2 ) š“š’Šš’=0 š“š’Šš’=-x šœšØš¬ šŸš…š’‡š’• + š“š’Šš’ + šœšØš¬ šŸš…š’‡š’• + š“ š“š’Šš’=0 š“š’Šš’=-x A non zero initial phase can change the radiation pattern
  19. Multiple antennas
  20. Tx1 Tx2 š’… šœƒ Tx(N-1) š’… . . . 2šœ‹ š‘‘š‘š‘œš‘ (šœƒ) šœ† Tx(N) Rx cos 2šœ‹š‘“š‘” + cos 2šœ‹š‘“š‘” + šœ™ + cos 2šœ‹š‘“š‘” + 2šœ™ + cos 2šœ‹š‘“š‘” + š‘ āˆ’ 2 āˆ— šœ™ + cos 2šœ‹š‘“š‘” + š‘ āˆ’ 1 āˆ— šœ™ š‘…š‘„ = ā€¦ā€¦..
  21. š‘…š‘„ = cos 2šœ‹š‘“š‘” + cos 2šœ‹š‘“š‘” + šœ™ + cos(2šœ‹š‘“š‘” + 2šœ™) + ā€¦ ā€¦ . . + cos 2šœ‹š‘“š‘” + š‘ āˆ’ 2 āˆ— šœ™ + cos(2šœ‹š‘“š‘” + š‘ āˆ’ 1 āˆ— šœ™) cos 2šœ‹š‘“š‘” = š‘’š‘–2šœ‹š‘“š‘” + š‘’āˆ’š‘–2šœ‹š‘“š‘” 2 = Re {ei2šœ‹š‘“š‘”} š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘”+šœ™ + ei2šœ‹š‘“š‘”+2šœ™ + ā€¦ ā€¦ . . ei2šœ‹š‘“š‘”+ š‘āˆ’1 šœ™ + ei2šœ‹š‘“š‘”+ š‘āˆ’1 šœ™} š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘”š‘’š‘–šœ™ + ei2šœ‹š‘“š‘”š‘’š‘–2šœ™ + ā€¦ ā€¦ . . ei2šœ‹š‘“š‘”š‘’š‘– š‘āˆ’2 šœ™ + ei2šœ‹š‘“š‘”š‘’š‘– š‘āˆ’1 šœ™} š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” 1 + š‘’š‘–šœ™ + š‘’š‘–2šœ™ + ā€¦ ā€¦ . . + š‘’š‘– š‘āˆ’2 šœ™ + š‘’š‘– š‘āˆ’1 šœ™ ) š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” 1 āˆ’ š‘’š‘–š‘šœ™ 1 āˆ’ š‘’š‘–šœ™ } š‘¹š’™(šœ½) = š‘¹š’†{šžš¢šŸš…š’‡š’• šŸ āˆ’ š’† š’Šš‘µ šŸš…š’…š’„š’š’”(šœ½) š€ šŸ āˆ’ š’† š’Š šŸš…š’…š’„š’š’”(šœ½) š€ }
  22. Radiation pattern (š‘‘ = šœ† 2 ) (š‘ = 2) (š‘ = 4) (š‘ = 8)
  23. š‘…š‘„ = cos 2šœ‹š‘“š‘” + cos 2šœ‹š‘“š‘” + šœ™ + cos(2šœ‹š‘“š‘” + 2šœ™) + ā€¦ ā€¦ . . + cos 2šœ‹š‘“š‘” + š‘ āˆ’ 2 āˆ— šœ™ + cos(2šœ‹š‘“š‘” + š‘ āˆ’ 1 āˆ— šœ™) š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘”+šœ™+šœ™š‘–š‘› + ei2šœ‹š‘“š‘”+2šœ™+2šœ™š‘–š‘› + ā€¦ ā€¦ . . ei2šœ‹š‘“š‘”+ š‘āˆ’2 šœ™+(š‘āˆ’2)šœ™š‘–š‘› + ei2šœ‹š‘“š‘”+ š‘āˆ’1 šœ™+(š‘āˆ’2)šœ™š‘–š‘›} š‘…š‘„ = cos 2šœ‹š‘“š‘” + šœ™š‘–š‘›š‘œ + cos 2šœ‹š‘“š‘” + šœ™ + šœ™š‘–š‘›1 + cos(2šœ‹š‘“š‘” + 2šœ™ + šœ™š‘–š‘›2) + ā€¦ + cos 2šœ‹š‘“š‘” + š‘ āˆ’ 2 āˆ— šœ™ + šœ™š‘–š‘›(š‘āˆ’2) + cos(2šœ‹š‘“š‘” + š‘ āˆ’ 1 āˆ— šœ™ + šœ™š‘–š‘›(š‘āˆ’1)) šœ™š‘–š‘›š‘œ = 0, šœ™š‘–š‘›1 = šœ™š‘–š‘› , šœ™š‘–š‘›2 = 2šœ™š‘–š‘› ā€¦ ā€¦ ā€¦ . . , šœ™š‘–š‘›1 = (š‘ āˆ’ 1) āˆ— šœ™š‘–š‘› šœ™ = 2šœ‹ šœ† āˆ— š’…š’„š’š’”(šœ½) š‘†š‘’š‘” šœ™š‘–š‘› = āˆ’šœ™ = āˆ’ 2šœ‹ šœ† āˆ— š’…š’„š’š’”(šœ½) š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘” + ā€¦ ā€¦ . . ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘”} š‘…š‘„ = š‘…š‘’{Nei2šœ‹š‘“š‘”} A maxima occurs in the direction of šœ½ Rotating the beam š‘…š‘„ = š‘…š‘’{ei2šœ‹š‘“š‘” + ei2šœ‹š‘“š‘”+šœ™+šœ™š‘–š‘›0 + ei2šœ‹š‘“š‘”+2šœ™+2šœ™š‘–š‘›1 + ā€¦ ā€¦ . . ei2šœ‹š‘“š‘”+ š‘āˆ’2 šœ™+šœ™š‘–š‘›(š‘āˆ’2) + ei2šœ‹š‘“š‘”+ š‘āˆ’1 šœ™+šœ™š‘–š‘›(š‘āˆ’1}
  24. š“š’Šš’ = āˆ’š“ = āˆ’ šŸš… š€ āˆ— š’…š’„š’š’”(šŸ’šŸ“) š“š’Šš’ = āˆ’š“ = āˆ’ šŸš… š€ āˆ— š’…š’„š’š’”(šŸ”šŸŽ) Rotating the beam
  25. Networking applications 25
  26. Acoustic Beamforming ā€“ noise suppression Silent zone Audible Zone
  27. Other applications ā€¢ Localization ā€¢ Gesture tracking ā€¢ RF Imaging
  28. Reception
  29. Sensing Angle of Arrival (AoA) Rx2 Rx1 š’… Path difference = šœƒ š’…š’„š’š’”(šœ½) šœ™ = 2šœ‹ šœ† āˆ— š’…š’„š’š’”(šœ½) šœšØš¬ šŸš…š’‡š’• + š“ Tx šœšØš¬ šŸš…š’‡š’• šœ½(š‘Øš’š‘Ø) = š’‚š’„š’š’” š€š“ šŸš…š’…
  30. Rx1 Rx2 š’… šœƒ Rx(N-1) š’… . . . 2šœ‹ š‘‘š‘š‘œš‘ (šœƒ) šœ† Rx(N) Tx cos 2šœ‹š‘“š‘” cos 2šœ‹š‘“š‘” + šœ™ cos(2šœ‹š‘“š‘” + š‘ āˆ’ 1 āˆ— šœ™) Antenna array
  31. cos 2šœ‹š‘“š‘” cos 2šœ‹š‘“š‘” + šœ™ cos 2šœ‹š‘“š‘” + 2šœ™ cos 2šœ‹š‘“š‘” + (š‘ āˆ’ 1)šœ™ cos 2šœ‹š‘“š‘” + (š‘ āˆ’ 2)šœ™ š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 š‘’š‘–2šœ‹š‘“š‘” š‘’š‘–2šœ‹š‘“š‘”+šœ™ š‘’š‘–2šœ‹š‘“š‘”+2šœ™ š‘’š‘–2šœ‹š‘“š‘”+(š‘āˆ’1)šœ™ š‘’š‘–2šœ‹š‘“š‘”+(š‘āˆ’2)šœ™ š‘’š‘–0 š‘’š‘–šœ™ š‘’š‘–2šœ™ š‘’š‘–šœ™ š‘’š‘–(š‘āˆ’2)šœ™ š‘’š‘–2šœ‹š‘“š‘” = = = š‘’š‘–0 š‘’š‘–šœ™ š‘’š‘–2šœ™ š‘’š‘–šœ™ š‘’š‘–(š‘āˆ’2)šœ™ š‘ š‘” =
  32. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 = š‘’š‘–0 š‘’š‘–šœ™ š‘’š‘–2šœ™ š‘’š‘–šœ™ š‘’š‘–(š‘āˆ’2)šœ™ Steering vector š‘ š‘” 2šœ‹ š‘‘š‘š‘œš‘ (šœƒ) šœ†
  33. Rx1 Rx2 š’… šœƒ Rx(N-1) š’… . . . Rx(N) Tx1 Tx2 Multiple transmitters
  34. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 š‘’š‘–0 š‘’š‘–šœ™1 š‘’š‘–2šœ™1 š‘’š‘–(š‘āˆ’1)šœ™1 š‘’š‘– š‘āˆ’2 šœ™1 š‘ 1 = š‘’š‘–0 š‘’š‘–šœ™2 š‘’š‘–2šœ™2 š‘’š‘–(š‘āˆ’1)šœ™2 š‘’š‘– š‘āˆ’2 šœ™2 š‘ 2 + š‘’š‘–0 š‘’š‘–šœ™š‘˜ š‘’š‘–2šœ™š‘˜ š‘’š‘–(š‘āˆ’1)šœ™š‘˜ š‘’š‘– š‘āˆ’2 šœ™š‘˜ š‘ š‘˜ + 2šœ‹ š‘‘š‘š‘œš‘ (šœƒ1) šœ† 2šœ‹ š‘‘š‘š‘œš‘ (šœƒ2) šœ† 2šœ‹ š‘‘š‘š‘œš‘ (šœƒš‘˜) šœ† Multiple transmitters Output is a linear combination of steering vectors from different directions
  35. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 š‘’š‘–0 š‘’š‘–šœ™1 š‘’š‘–2šœ™1 š‘’š‘–(š‘āˆ’1)šœ™1 š‘’š‘– š‘āˆ’2 šœ™1 š‘ 1 = š‘’š‘–0 š‘’š‘–šœ™2 š‘’š‘–2šœ™2 š‘’š‘–(š‘āˆ’1)šœ™2 š‘’š‘– š‘āˆ’2 šœ™2 š‘ 2 š‘’š‘–0 š‘’š‘–šœ™š‘˜ š‘’š‘–2šœ™š‘˜ š‘’š‘–(š‘āˆ’1)šœ™š‘˜ š‘’š‘– š‘āˆ’2 šœ™š‘˜ š‘ š‘˜ K sources (Input Vector) N receivers (Output vector) Steering Matrix (N x K) Multiple transmitters
  36. Detecting AoA of K sources simultaneously
  37. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 š‘’š‘–0 š‘’š‘–šœ™1 š‘’š‘–2šœ™1 š‘’š‘–(š‘āˆ’1)šœ™1 š‘’š‘– š‘āˆ’2 šœ™1 š‘ 1 = š‘’š‘–0 š‘’š‘–šœ™2 š‘’š‘–2šœ™2 š‘’š‘–(š‘āˆ’1)šœ™2 š‘’š‘– š‘āˆ’2 šœ™2 š‘ 2 š‘’š‘–0 š‘’š‘–šœ™š‘˜ š‘’š‘–2šœ™š‘˜ š‘’š‘–(š‘āˆ’1)šœ™š‘˜ š‘’š‘– š‘āˆ’2 šœ™š‘˜ š‘ š‘˜
  38. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 š‘’š‘–0 š‘’š‘–šœ™1 š‘’š‘–2šœ™1 š‘’š‘–(š‘āˆ’1)šœ™1 š‘’š‘– š‘āˆ’2 šœ™1 š‘ 1 = š‘’š‘–0 š‘’š‘–šœ™2 š‘’š‘–2šœ™2 š‘’š‘–(š‘āˆ’1)šœ™2 š‘’š‘– š‘āˆ’2 šœ™2 š‘ 2 š‘’š‘–0 š‘’š‘–šœ™š‘˜ š‘’š‘–2šœ™š‘˜ š‘’š‘–(š‘āˆ’1)šœ™š‘˜ š‘’š‘– š‘āˆ’2 šœ™š‘˜ š‘ š‘˜ Multiply by conjugate of steering vector of source 1 š‘’āˆ’š‘–(š‘āˆ’1)šœ™1 š‘’š‘–0 š‘’āˆ’š‘–šœ™1 š‘’āˆ’š‘–2šœ™1 .. š‘’āˆ’š‘–(š‘āˆ’1)šœ™1 š‘’š‘–0 š‘’āˆ’š‘–šœ™1 š‘’āˆ’š‘–2šœ™1 ..
  39. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 š‘ 1 = š‘ 2 š‘ š‘˜ š‘ š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ š‘’āˆ’š‘–(š‘āˆ’1)šœ™1 š‘’š‘–0 š‘’āˆ’š‘–šœ™1 š‘’āˆ’š‘–2šœ™1 ..
  40. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 = š‘ 1 āˆ— š‘ + š‘ 2 āˆ— š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ + š‘ 3 āˆ— š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ + ā€¦ ā€¦ . . All energy from direction šœƒ1(š‘“š‘Ÿš‘œš‘š š‘ 1) have been aggregated and amplified A(šœƒ1) = š‘’āˆ’š‘–(š‘āˆ’1)šœ™1 š‘’š‘–0 š‘’āˆ’š‘–šœ™1 š‘’āˆ’š‘–2šœ™1 ..
  41. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 = š‘ 1 āˆ— (š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’) + š‘ 2 āˆ— š‘ + š‘ 3 āˆ— š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ + ā€¦ ā€¦ . . All energy from direction šœƒ2(š‘“š‘Ÿš‘œš‘š š‘ 2) have been aggregated and amplified A(šœƒ2) = š‘’āˆ’š‘–(š‘āˆ’1)šœ™2 š‘’š‘–0 š‘’āˆ’š‘–šœ™2 š‘’āˆ’š‘–2šœ™2..
  42. š‘…š‘„1 š‘…š‘„2 š‘…š‘„3 š‘…š‘„š‘ š‘…š‘„š‘āˆ’1 = š‘ 1 āˆ— (š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’) + š‘ 2 āˆ— š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ + š‘ 3 āˆ— š‘ š‘šš‘Žš‘™š‘™ š‘£š‘Žš‘™š‘¢š‘’ + ā€¦ ā€¦ . . The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals A(šœƒš‘Ÿ) = š‘’āˆ’š‘–(š‘āˆ’1)šœ™š‘Ÿ š‘’š‘–0 š‘’āˆ’š‘–šœ™š‘Ÿ š‘’āˆ’š‘–2šœ™š‘Ÿ..
  43. ā€¢ Construct a graph of for all values of ā€¢ Any active source from direction should have a peak in the above graph .. ā€¢ This is called delay and sum beamforming A(šœƒ) šœƒ šœƒš‘ 
  44. Detecting multiple AoA Suc A(šœƒ) š‘»š’™šŸ š‘»š’™šŸ š‘»š’™šŸ‘ AoA Spectrum
  45. Close by AoAs cannot be resolved š‘»š’™šŸ š‘»š’™šŸ š‘»š’™šŸ‘
  46. MUSIC algorithm has sharp peaks to resolve close AoA Based on eigen decomposition and PCA ā€“ reference to be provided š“š‘šš‘¢š‘ š‘–š‘(šœƒ) š‘»š’™šŸ š‘»š’™šŸ š‘»š’™šŸ‘
  47. Degrees of freedom for beamforming ā€¢ Antenna separation ā€¢ Initial phases of antenna sources ā€¢ Number of antennas