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beamforming.pptx

  1. Beamforming
  2. Tx1
  3. Tx1 cos(2๐œ‹๐‘“๐‘ก)
  4. Tx1 Tx2 ๐€ ๐Ÿ cos(2๐œ‹๐‘“๐‘ก)
  5. Tx1 Tx2 ๐€ ๐Ÿ cos(2๐œ‹๐‘“๐‘ก) Rx ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐…
  6. Destructive superimposition Tx1 Tx2 ๐€ ๐Ÿ Zero signal cos 2๐œ‹๐‘“๐‘ก + cos 2๐œ‹๐‘“๐‘ก + ๐œ‹ = 0
  7. Tx1 Tx2 ๐€ ๐Ÿ Rx ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’•
  8. Constructive superimposition Tx1 Tx2 ๐€ ๐Ÿ Amplified signal (twice amplitude) cos 2๐œ‹๐‘“๐‘ก + cos 2๐œ‹๐‘“๐‘ก + 0 = 2cos(2๐œ‹๐‘“๐‘ก)
  9. Receiver at arbitrary location Tx1 Tx2 ๐€ ๐Ÿ ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“ Rx
  10. Arbitrary location, whatโ€™s the path difference Path difference = ??
  11. Path difference Tx1 Tx2 ๐’… ๐œƒ ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“ Rx
  12. Path difference and phase difference Tx1 Tx2 ๐’… Path difference = ๐œƒ ๐’…๐’„๐’๐’”(๐œฝ) ๐œ™(๐‘โ„Ž๐‘Ž๐‘ ๐‘’ ๐‘‘๐‘–๐‘“๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’) = 2๐œ‹ ๐œ† โˆ— (๐‘๐‘Ž๐‘กโ„Ž ๐‘‘๐‘–๐‘“๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’) ๐œ™ = 2๐œ‹ ๐œ† โˆ— ๐’…๐’„๐’๐’”(๐œฝ) ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“ Rx ๐‘๐ฑ ๐œฝ = ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐Ÿ๐… ๐€ ๐’…๐’„๐’๐’”(๐œฝ)
  13. (๐‘‘ = ๐œ† 2 ) Radiation pattern: Rx amplitude as a function of angle
  14. (๐‘‘ = ๐œ†) Radiation pattern: Rx amplitude as a function of angle
  15. Radiation pattern: Rx amplitude as a function of angle (๐‘‘ = 2๐œ†)
  16. (๐‘‘ = ๐œ† 2 ) Radiation pattern: Rx amplitude as a function of angle ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“
  17. (๐‘‘ = ๐œ† 2 ) Radiation pattern: Rx amplitude as a function of angle ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“๐’Š๐’ + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“ The initial phases can be controlled
  18. Radiation pattern: Rx amplitude as a function of angle (๐‘‘ = ๐œ† 2 ) ๐“๐’Š๐’=0 ๐“๐’Š๐’=-x ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“๐’Š๐’ + ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“ ๐“๐’Š๐’=0 ๐“๐’Š๐’=-x A non zero initial phase can change the radiation pattern
  19. Multiple antennas
  20. Tx1 Tx2 ๐’… ๐œƒ Tx(N-1) ๐’… . . . 2๐œ‹ ๐‘‘๐‘๐‘œ๐‘ (๐œƒ) ๐œ† Tx(N) Rx cos 2๐œ‹๐‘“๐‘ก + cos 2๐œ‹๐‘“๐‘ก + ๐œ™ + cos 2๐œ‹๐‘“๐‘ก + 2๐œ™ + cos 2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 2 โˆ— ๐œ™ + cos 2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 1 โˆ— ๐œ™ ๐‘…๐‘ฅ = โ€ฆโ€ฆ..
  21. ๐‘…๐‘ฅ = cos 2๐œ‹๐‘“๐‘ก + cos 2๐œ‹๐‘“๐‘ก + ๐œ™ + cos(2๐œ‹๐‘“๐‘ก + 2๐œ™) + โ€ฆ โ€ฆ . . + cos 2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 2 โˆ— ๐œ™ + cos(2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 1 โˆ— ๐œ™) cos 2๐œ‹๐‘“๐‘ก = ๐‘’๐‘–2๐œ‹๐‘“๐‘ก + ๐‘’โˆ’๐‘–2๐œ‹๐‘“๐‘ก 2 = Re {ei2๐œ‹๐‘“๐‘ก} ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก+๐œ™ + ei2๐œ‹๐‘“๐‘ก+2๐œ™ + โ€ฆ โ€ฆ . . ei2๐œ‹๐‘“๐‘ก+ ๐‘โˆ’1 ๐œ™ + ei2๐œ‹๐‘“๐‘ก+ ๐‘โˆ’1 ๐œ™} ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก๐‘’๐‘–๐œ™ + ei2๐œ‹๐‘“๐‘ก๐‘’๐‘–2๐œ™ + โ€ฆ โ€ฆ . . ei2๐œ‹๐‘“๐‘ก๐‘’๐‘– ๐‘โˆ’2 ๐œ™ + ei2๐œ‹๐‘“๐‘ก๐‘’๐‘– ๐‘โˆ’1 ๐œ™} ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก 1 + ๐‘’๐‘–๐œ™ + ๐‘’๐‘–2๐œ™ + โ€ฆ โ€ฆ . . + ๐‘’๐‘– ๐‘โˆ’2 ๐œ™ + ๐‘’๐‘– ๐‘โˆ’1 ๐œ™ ) ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก 1 โˆ’ ๐‘’๐‘–๐‘๐œ™ 1 โˆ’ ๐‘’๐‘–๐œ™ } ๐‘น๐’™(๐œฝ) = ๐‘น๐’†{๐ž๐ข๐Ÿ๐…๐’‡๐’• ๐Ÿ โˆ’ ๐’† ๐’Š๐‘ต ๐Ÿ๐…๐’…๐’„๐’๐’”(๐œฝ) ๐€ ๐Ÿ โˆ’ ๐’† ๐’Š ๐Ÿ๐…๐’…๐’„๐’๐’”(๐œฝ) ๐€ }
  22. Radiation pattern (๐‘‘ = ๐œ† 2 ) (๐‘ = 2) (๐‘ = 4) (๐‘ = 8)
  23. ๐‘…๐‘ฅ = cos 2๐œ‹๐‘“๐‘ก + cos 2๐œ‹๐‘“๐‘ก + ๐œ™ + cos(2๐œ‹๐‘“๐‘ก + 2๐œ™) + โ€ฆ โ€ฆ . . + cos 2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 2 โˆ— ๐œ™ + cos(2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 1 โˆ— ๐œ™) ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก+๐œ™+๐œ™๐‘–๐‘› + ei2๐œ‹๐‘“๐‘ก+2๐œ™+2๐œ™๐‘–๐‘› + โ€ฆ โ€ฆ . . ei2๐œ‹๐‘“๐‘ก+ ๐‘โˆ’2 ๐œ™+(๐‘โˆ’2)๐œ™๐‘–๐‘› + ei2๐œ‹๐‘“๐‘ก+ ๐‘โˆ’1 ๐œ™+(๐‘โˆ’2)๐œ™๐‘–๐‘›} ๐‘…๐‘ฅ = cos 2๐œ‹๐‘“๐‘ก + ๐œ™๐‘–๐‘›๐‘œ + cos 2๐œ‹๐‘“๐‘ก + ๐œ™ + ๐œ™๐‘–๐‘›1 + cos(2๐œ‹๐‘“๐‘ก + 2๐œ™ + ๐œ™๐‘–๐‘›2) + โ€ฆ + cos 2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 2 โˆ— ๐œ™ + ๐œ™๐‘–๐‘›(๐‘โˆ’2) + cos(2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 1 โˆ— ๐œ™ + ๐œ™๐‘–๐‘›(๐‘โˆ’1)) ๐œ™๐‘–๐‘›๐‘œ = 0, ๐œ™๐‘–๐‘›1 = ๐œ™๐‘–๐‘› , ๐œ™๐‘–๐‘›2 = 2๐œ™๐‘–๐‘› โ€ฆ โ€ฆ โ€ฆ . . , ๐œ™๐‘–๐‘›1 = (๐‘ โˆ’ 1) โˆ— ๐œ™๐‘–๐‘› ๐œ™ = 2๐œ‹ ๐œ† โˆ— ๐’…๐’„๐’๐’”(๐œฝ) ๐‘†๐‘’๐‘ก ๐œ™๐‘–๐‘› = โˆ’๐œ™ = โˆ’ 2๐œ‹ ๐œ† โˆ— ๐’…๐’„๐’๐’”(๐œฝ) ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก + โ€ฆ โ€ฆ . . ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก} ๐‘…๐‘ฅ = ๐‘…๐‘’{Nei2๐œ‹๐‘“๐‘ก} A maxima occurs in the direction of ๐œฝ Rotating the beam ๐‘…๐‘ฅ = ๐‘…๐‘’{ei2๐œ‹๐‘“๐‘ก + ei2๐œ‹๐‘“๐‘ก+๐œ™+๐œ™๐‘–๐‘›0 + ei2๐œ‹๐‘“๐‘ก+2๐œ™+2๐œ™๐‘–๐‘›1 + โ€ฆ โ€ฆ . . ei2๐œ‹๐‘“๐‘ก+ ๐‘โˆ’2 ๐œ™+๐œ™๐‘–๐‘›(๐‘โˆ’2) + ei2๐œ‹๐‘“๐‘ก+ ๐‘โˆ’1 ๐œ™+๐œ™๐‘–๐‘›(๐‘โˆ’1}
  24. ๐“๐’Š๐’ = โˆ’๐“ = โˆ’ ๐Ÿ๐… ๐€ โˆ— ๐’…๐’„๐’๐’”(๐Ÿ’๐Ÿ“) ๐“๐’Š๐’ = โˆ’๐“ = โˆ’ ๐Ÿ๐… ๐€ โˆ— ๐’…๐’„๐’๐’”(๐Ÿ”๐ŸŽ) Rotating the beam
  25. Networking applications 25
  26. Acoustic Beamforming โ€“ noise suppression Silent zone Audible Zone
  27. Other applications โ€ข Localization โ€ข Gesture tracking โ€ข RF Imaging
  28. Reception
  29. Sensing Angle of Arrival (AoA) Rx2 Rx1 ๐’… Path difference = ๐œƒ ๐’…๐’„๐’๐’”(๐œฝ) ๐œ™ = 2๐œ‹ ๐œ† โˆ— ๐’…๐’„๐’๐’”(๐œฝ) ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• + ๐“ Tx ๐œ๐จ๐ฌ ๐Ÿ๐…๐’‡๐’• ๐œฝ(๐‘จ๐’๐‘จ) = ๐’‚๐’„๐’๐’” ๐€๐“ ๐Ÿ๐…๐’…
  30. Rx1 Rx2 ๐’… ๐œƒ Rx(N-1) ๐’… . . . 2๐œ‹ ๐‘‘๐‘๐‘œ๐‘ (๐œƒ) ๐œ† Rx(N) Tx cos 2๐œ‹๐‘“๐‘ก cos 2๐œ‹๐‘“๐‘ก + ๐œ™ cos(2๐œ‹๐‘“๐‘ก + ๐‘ โˆ’ 1 โˆ— ๐œ™) Antenna array
  31. cos 2๐œ‹๐‘“๐‘ก cos 2๐œ‹๐‘“๐‘ก + ๐œ™ cos 2๐œ‹๐‘“๐‘ก + 2๐œ™ cos 2๐œ‹๐‘“๐‘ก + (๐‘ โˆ’ 1)๐œ™ cos 2๐œ‹๐‘“๐‘ก + (๐‘ โˆ’ 2)๐œ™ ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 ๐‘’๐‘–2๐œ‹๐‘“๐‘ก ๐‘’๐‘–2๐œ‹๐‘“๐‘ก+๐œ™ ๐‘’๐‘–2๐œ‹๐‘“๐‘ก+2๐œ™ ๐‘’๐‘–2๐œ‹๐‘“๐‘ก+(๐‘โˆ’1)๐œ™ ๐‘’๐‘–2๐œ‹๐‘“๐‘ก+(๐‘โˆ’2)๐œ™ ๐‘’๐‘–0 ๐‘’๐‘–๐œ™ ๐‘’๐‘–2๐œ™ ๐‘’๐‘–๐œ™ ๐‘’๐‘–(๐‘โˆ’2)๐œ™ ๐‘’๐‘–2๐œ‹๐‘“๐‘ก = = = ๐‘’๐‘–0 ๐‘’๐‘–๐œ™ ๐‘’๐‘–2๐œ™ ๐‘’๐‘–๐œ™ ๐‘’๐‘–(๐‘โˆ’2)๐œ™ ๐‘ ๐‘ก =
  32. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 = ๐‘’๐‘–0 ๐‘’๐‘–๐œ™ ๐‘’๐‘–2๐œ™ ๐‘’๐‘–๐œ™ ๐‘’๐‘–(๐‘โˆ’2)๐œ™ Steering vector ๐‘ ๐‘ก 2๐œ‹ ๐‘‘๐‘๐‘œ๐‘ (๐œƒ) ๐œ†
  33. Rx1 Rx2 ๐’… ๐œƒ Rx(N-1) ๐’… . . . Rx(N) Tx1 Tx2 Multiple transmitters
  34. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™1 ๐‘’๐‘–2๐œ™1 ๐‘’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™1 ๐‘ 1 = ๐‘’๐‘–0 ๐‘’๐‘–๐œ™2 ๐‘’๐‘–2๐œ™2 ๐‘’๐‘–(๐‘โˆ’1)๐œ™2 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™2 ๐‘ 2 + ๐‘’๐‘–0 ๐‘’๐‘–๐œ™๐‘˜ ๐‘’๐‘–2๐œ™๐‘˜ ๐‘’๐‘–(๐‘โˆ’1)๐œ™๐‘˜ ๐‘’๐‘– ๐‘โˆ’2 ๐œ™๐‘˜ ๐‘ ๐‘˜ + 2๐œ‹ ๐‘‘๐‘๐‘œ๐‘ (๐œƒ1) ๐œ† 2๐œ‹ ๐‘‘๐‘๐‘œ๐‘ (๐œƒ2) ๐œ† 2๐œ‹ ๐‘‘๐‘๐‘œ๐‘ (๐œƒ๐‘˜) ๐œ† Multiple transmitters Output is a linear combination of steering vectors from different directions
  35. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™1 ๐‘’๐‘–2๐œ™1 ๐‘’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™1 ๐‘ 1 = ๐‘’๐‘–0 ๐‘’๐‘–๐œ™2 ๐‘’๐‘–2๐œ™2 ๐‘’๐‘–(๐‘โˆ’1)๐œ™2 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™2 ๐‘ 2 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™๐‘˜ ๐‘’๐‘–2๐œ™๐‘˜ ๐‘’๐‘–(๐‘โˆ’1)๐œ™๐‘˜ ๐‘’๐‘– ๐‘โˆ’2 ๐œ™๐‘˜ ๐‘ ๐‘˜ K sources (Input Vector) N receivers (Output vector) Steering Matrix (N x K) Multiple transmitters
  36. Detecting AoA of K sources simultaneously
  37. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™1 ๐‘’๐‘–2๐œ™1 ๐‘’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™1 ๐‘ 1 = ๐‘’๐‘–0 ๐‘’๐‘–๐œ™2 ๐‘’๐‘–2๐œ™2 ๐‘’๐‘–(๐‘โˆ’1)๐œ™2 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™2 ๐‘ 2 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™๐‘˜ ๐‘’๐‘–2๐œ™๐‘˜ ๐‘’๐‘–(๐‘โˆ’1)๐œ™๐‘˜ ๐‘’๐‘– ๐‘โˆ’2 ๐œ™๐‘˜ ๐‘ ๐‘˜
  38. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™1 ๐‘’๐‘–2๐œ™1 ๐‘’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™1 ๐‘ 1 = ๐‘’๐‘–0 ๐‘’๐‘–๐œ™2 ๐‘’๐‘–2๐œ™2 ๐‘’๐‘–(๐‘โˆ’1)๐œ™2 ๐‘’๐‘– ๐‘โˆ’2 ๐œ™2 ๐‘ 2 ๐‘’๐‘–0 ๐‘’๐‘–๐œ™๐‘˜ ๐‘’๐‘–2๐œ™๐‘˜ ๐‘’๐‘–(๐‘โˆ’1)๐œ™๐‘˜ ๐‘’๐‘– ๐‘โˆ’2 ๐œ™๐‘˜ ๐‘ ๐‘˜ Multiply by conjugate of steering vector of source 1 ๐‘’โˆ’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘–0 ๐‘’โˆ’๐‘–๐œ™1 ๐‘’โˆ’๐‘–2๐œ™1 .. ๐‘’โˆ’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘–0 ๐‘’โˆ’๐‘–๐œ™1 ๐‘’โˆ’๐‘–2๐œ™1 ..
  39. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 ๐‘ 1 = ๐‘ 2 ๐‘ ๐‘˜ ๐‘ ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ ๐‘’โˆ’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘–0 ๐‘’โˆ’๐‘–๐œ™1 ๐‘’โˆ’๐‘–2๐œ™1 ..
  40. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 = ๐‘ 1 โˆ— ๐‘ + ๐‘ 2 โˆ— ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ + ๐‘ 3 โˆ— ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ + โ€ฆ โ€ฆ . . All energy from direction ๐œƒ1(๐‘“๐‘Ÿ๐‘œ๐‘š ๐‘ 1) have been aggregated and amplified A(๐œƒ1) = ๐‘’โˆ’๐‘–(๐‘โˆ’1)๐œ™1 ๐‘’๐‘–0 ๐‘’โˆ’๐‘–๐œ™1 ๐‘’โˆ’๐‘–2๐œ™1 ..
  41. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 = ๐‘ 1 โˆ— (๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’) + ๐‘ 2 โˆ— ๐‘ + ๐‘ 3 โˆ— ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ + โ€ฆ โ€ฆ . . All energy from direction ๐œƒ2(๐‘“๐‘Ÿ๐‘œ๐‘š ๐‘ 2) have been aggregated and amplified A(๐œƒ2) = ๐‘’โˆ’๐‘–(๐‘โˆ’1)๐œ™2 ๐‘’๐‘–0 ๐‘’โˆ’๐‘–๐œ™2 ๐‘’โˆ’๐‘–2๐œ™2..
  42. ๐‘…๐‘ฅ1 ๐‘…๐‘ฅ2 ๐‘…๐‘ฅ3 ๐‘…๐‘ฅ๐‘ ๐‘…๐‘ฅ๐‘โˆ’1 = ๐‘ 1 โˆ— (๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’) + ๐‘ 2 โˆ— ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ + ๐‘ 3 โˆ— ๐‘ ๐‘š๐‘Ž๐‘™๐‘™ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ + โ€ฆ โ€ฆ . . The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals A(๐œƒ๐‘Ÿ) = ๐‘’โˆ’๐‘–(๐‘โˆ’1)๐œ™๐‘Ÿ ๐‘’๐‘–0 ๐‘’โˆ’๐‘–๐œ™๐‘Ÿ ๐‘’โˆ’๐‘–2๐œ™๐‘Ÿ..
  43. โ€ข Construct a graph of for all values of โ€ข Any active source from direction should have a peak in the above graph .. โ€ข This is called delay and sum beamforming A(๐œƒ) ๐œƒ ๐œƒ๐‘ 
  44. Detecting multiple AoA Suc A(๐œƒ) ๐‘ป๐’™๐Ÿ ๐‘ป๐’™๐Ÿ ๐‘ป๐’™๐Ÿ‘ AoA Spectrum
  45. Close by AoAs cannot be resolved ๐‘ป๐’™๐Ÿ ๐‘ป๐’™๐Ÿ ๐‘ป๐’™๐Ÿ‘
  46. MUSIC algorithm has sharp peaks to resolve close AoA Based on eigen decomposition and PCA โ€“ reference to be provided ๐ด๐‘š๐‘ข๐‘ ๐‘–๐‘(๐œƒ) ๐‘ป๐’™๐Ÿ ๐‘ป๐’™๐Ÿ ๐‘ป๐’™๐Ÿ‘
  47. Degrees of freedom for beamforming โ€ข Antenna separation โ€ข Initial phases of antenna sources โ€ข Number of antennas
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