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beamforming.pptx
- 1. Beamforming
- 2. Tx1
- 5. Tx1 Tx2 𝝀 𝟐 cos(2𝜋𝑓𝑡) Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝅
- 6. Destructive superimposition Tx1 Tx2 𝝀 𝟐 Zero signal cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡 + 𝜋 = 0
- 7. Tx1 Tx2 𝝀 𝟐 Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕
- 8. Constructive superimposition Tx1 Tx2 𝝀 𝟐 Amplified signal (twice amplitude) cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡 + 0 = 2cos(2𝜋𝑓𝑡)
- 9. Receiver at arbitrary location Tx1 Tx2 𝝀 𝟐 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓 Rx
- 10. Arbitrary location, what’s the path difference Path difference = ??
- 11. Path difference Tx1 Tx2 𝒅 𝜃 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓 Rx
- 12. Path difference and phase difference Tx1 Tx2 𝒅 Path difference = 𝜃 𝒅𝒄𝒐𝒔(𝜽) 𝜙(𝑝ℎ𝑎𝑠𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒) = 2𝜋 𝜆 ∗ (𝑝𝑎𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒) 𝜙 = 2𝜋 𝜆 ∗ 𝒅𝒄𝒐𝒔(𝜽) 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓 Rx 𝐑𝐱 𝜽 = 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝟐𝝅 𝝀 𝒅𝒄𝒐𝒔(𝜽)
- 13. (𝑑 = 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle
- 14. (𝑑 = 𝜆) Radiation pattern: Rx amplitude as a function of angle
- 15. Radiation pattern: Rx amplitude as a function of angle (𝑑 = 2𝜆)
- 16. (𝑑 = 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓
- 17. (𝑑 = 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓𝒊𝒏 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓 The initial phases can be controlled
- 18. Radiation pattern: Rx amplitude as a function of angle (𝑑 = 𝜆 2 ) 𝝓𝒊𝒏=0 𝝓𝒊𝒏=-x 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓𝒊𝒏 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓 𝝓𝒊𝒏=0 𝝓𝒊𝒏=-x A non zero initial phase can change the radiation pattern
- 20. Tx1 Tx2 𝒅 𝜃 Tx(N-1) 𝒅 . . . 2𝜋 𝑑𝑐𝑜𝑠(𝜃) 𝜆 Tx(N) Rx cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡 + 𝜙 + cos 2𝜋𝑓𝑡 + 2𝜙 + cos 2𝜋𝑓𝑡 + 𝑁 − 2 ∗ 𝜙 + cos 2𝜋𝑓𝑡 + 𝑁 − 1 ∗ 𝜙 𝑅𝑥 = ……..
- 21. 𝑅𝑥 = cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡 + 𝜙 + cos(2𝜋𝑓𝑡 + 2𝜙) + … … . . + cos 2𝜋𝑓𝑡 + 𝑁 − 2 ∗ 𝜙 + cos(2𝜋𝑓𝑡 + 𝑁 − 1 ∗ 𝜙) cos 2𝜋𝑓𝑡 = 𝑒𝑖2𝜋𝑓𝑡 + 𝑒−𝑖2𝜋𝑓𝑡 2 = Re {ei2𝜋𝑓𝑡} 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡+𝜙 + ei2𝜋𝑓𝑡+2𝜙 + … … . . ei2𝜋𝑓𝑡+ 𝑁−1 𝜙 + ei2𝜋𝑓𝑡+ 𝑁−1 𝜙} 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡𝑒𝑖𝜙 + ei2𝜋𝑓𝑡𝑒𝑖2𝜙 + … … . . ei2𝜋𝑓𝑡𝑒𝑖 𝑁−2 𝜙 + ei2𝜋𝑓𝑡𝑒𝑖 𝑁−1 𝜙} 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 1 + 𝑒𝑖𝜙 + 𝑒𝑖2𝜙 + … … . . + 𝑒𝑖 𝑁−2 𝜙 + 𝑒𝑖 𝑁−1 𝜙 ) 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 1 − 𝑒𝑖𝑁𝜙 1 − 𝑒𝑖𝜙 } 𝑹𝒙(𝜽) = 𝑹𝒆{𝐞𝐢𝟐𝝅𝒇𝒕 𝟏 − 𝒆 𝒊𝑵 𝟐𝝅𝒅𝒄𝒐𝒔(𝜽) 𝝀 𝟏 − 𝒆 𝒊 𝟐𝝅𝒅𝒄𝒐𝒔(𝜽) 𝝀 }
- 22. Radiation pattern (𝑑 = 𝜆 2 ) (𝑁 = 2) (𝑁 = 4) (𝑁 = 8)
- 23. 𝑅𝑥 = cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡 + 𝜙 + cos(2𝜋𝑓𝑡 + 2𝜙) + … … . . + cos 2𝜋𝑓𝑡 + 𝑁 − 2 ∗ 𝜙 + cos(2𝜋𝑓𝑡 + 𝑁 − 1 ∗ 𝜙) 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡+𝜙+𝜙𝑖𝑛 + ei2𝜋𝑓𝑡+2𝜙+2𝜙𝑖𝑛 + … … . . ei2𝜋𝑓𝑡+ 𝑁−2 𝜙+(𝑁−2)𝜙𝑖𝑛 + ei2𝜋𝑓𝑡+ 𝑁−1 𝜙+(𝑁−2)𝜙𝑖𝑛} 𝑅𝑥 = cos 2𝜋𝑓𝑡 + 𝜙𝑖𝑛𝑜 + cos 2𝜋𝑓𝑡 + 𝜙 + 𝜙𝑖𝑛1 + cos(2𝜋𝑓𝑡 + 2𝜙 + 𝜙𝑖𝑛2) + … + cos 2𝜋𝑓𝑡 + 𝑁 − 2 ∗ 𝜙 + 𝜙𝑖𝑛(𝑁−2) + cos(2𝜋𝑓𝑡 + 𝑁 − 1 ∗ 𝜙 + 𝜙𝑖𝑛(𝑁−1)) 𝜙𝑖𝑛𝑜 = 0, 𝜙𝑖𝑛1 = 𝜙𝑖𝑛 , 𝜙𝑖𝑛2 = 2𝜙𝑖𝑛 … … … . . , 𝜙𝑖𝑛1 = (𝑁 − 1) ∗ 𝜙𝑖𝑛 𝜙 = 2𝜋 𝜆 ∗ 𝒅𝒄𝒐𝒔(𝜽) 𝑆𝑒𝑡 𝜙𝑖𝑛 = −𝜙 = − 2𝜋 𝜆 ∗ 𝒅𝒄𝒐𝒔(𝜽) 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡 + … … . . ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡} 𝑅𝑥 = 𝑅𝑒{Nei2𝜋𝑓𝑡} A maxima occurs in the direction of 𝜽 Rotating the beam 𝑅𝑥 = 𝑅𝑒{ei2𝜋𝑓𝑡 + ei2𝜋𝑓𝑡+𝜙+𝜙𝑖𝑛0 + ei2𝜋𝑓𝑡+2𝜙+2𝜙𝑖𝑛1 + … … . . ei2𝜋𝑓𝑡+ 𝑁−2 𝜙+𝜙𝑖𝑛(𝑁−2) + ei2𝜋𝑓𝑡+ 𝑁−1 𝜙+𝜙𝑖𝑛(𝑁−1}
- 24. 𝝓𝒊𝒏 = −𝝓 = − 𝟐𝝅 𝝀 ∗ 𝒅𝒄𝒐𝒔(𝟒𝟓) 𝝓𝒊𝒏 = −𝝓 = − 𝟐𝝅 𝝀 ∗ 𝒅𝒄𝒐𝒔(𝟔𝟎) Rotating the beam
- 26. Acoustic Beamforming – noise suppression Silent zone Audible Zone
- 27. Other applications • Localization • Gesture tracking • RF Imaging
- 28. Reception
- 29. Sensing Angle of Arrival (AoA) Rx2 Rx1 𝒅 Path difference = 𝜃 𝒅𝒄𝒐𝒔(𝜽) 𝜙 = 2𝜋 𝜆 ∗ 𝒅𝒄𝒐𝒔(𝜽) 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝝓 Tx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 𝜽(𝑨𝒐𝑨) = 𝒂𝒄𝒐𝒔 𝝀𝝓 𝟐𝝅𝒅
- 30. Rx1 Rx2 𝒅 𝜃 Rx(N-1) 𝒅 . . . 2𝜋 𝑑𝑐𝑜𝑠(𝜃) 𝜆 Rx(N) Tx cos 2𝜋𝑓𝑡 cos 2𝜋𝑓𝑡 + 𝜙 cos(2𝜋𝑓𝑡 + 𝑁 − 1 ∗ 𝜙) Antenna array
- 31. cos 2𝜋𝑓𝑡 cos 2𝜋𝑓𝑡 + 𝜙 cos 2𝜋𝑓𝑡 + 2𝜙 cos 2𝜋𝑓𝑡 + (𝑁 − 1)𝜙 cos 2𝜋𝑓𝑡 + (𝑁 − 2)𝜙 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 𝑒𝑖2𝜋𝑓𝑡 𝑒𝑖2𝜋𝑓𝑡+𝜙 𝑒𝑖2𝜋𝑓𝑡+2𝜙 𝑒𝑖2𝜋𝑓𝑡+(𝑁−1)𝜙 𝑒𝑖2𝜋𝑓𝑡+(𝑁−2)𝜙 𝑒𝑖0 𝑒𝑖𝜙 𝑒𝑖2𝜙 𝑒𝑖𝜙 𝑒𝑖(𝑁−2)𝜙 𝑒𝑖2𝜋𝑓𝑡 = = = 𝑒𝑖0 𝑒𝑖𝜙 𝑒𝑖2𝜙 𝑒𝑖𝜙 𝑒𝑖(𝑁−2)𝜙 𝑠𝑡 =
- 33. Rx1 Rx2 𝒅 𝜃 Rx(N-1) 𝒅 . . . Rx(N) Tx1 Tx2 Multiple transmitters
- 34. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 𝑒𝑖0 𝑒𝑖𝜙1 𝑒𝑖2𝜙1 𝑒𝑖(𝑁−1)𝜙1 𝑒𝑖 𝑁−2 𝜙1 𝑠1 = 𝑒𝑖0 𝑒𝑖𝜙2 𝑒𝑖2𝜙2 𝑒𝑖(𝑁−1)𝜙2 𝑒𝑖 𝑁−2 𝜙2 𝑠2 + 𝑒𝑖0 𝑒𝑖𝜙𝑘 𝑒𝑖2𝜙𝑘 𝑒𝑖(𝑁−1)𝜙𝑘 𝑒𝑖 𝑁−2 𝜙𝑘 𝑠𝑘 + 2𝜋 𝑑𝑐𝑜𝑠(𝜃1) 𝜆 2𝜋 𝑑𝑐𝑜𝑠(𝜃2) 𝜆 2𝜋 𝑑𝑐𝑜𝑠(𝜃𝑘) 𝜆 Multiple transmitters Output is a linear combination of steering vectors from different directions
- 35. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 𝑒𝑖0 𝑒𝑖𝜙1 𝑒𝑖2𝜙1 𝑒𝑖(𝑁−1)𝜙1 𝑒𝑖 𝑁−2 𝜙1 𝑠1 = 𝑒𝑖0 𝑒𝑖𝜙2 𝑒𝑖2𝜙2 𝑒𝑖(𝑁−1)𝜙2 𝑒𝑖 𝑁−2 𝜙2 𝑠2 𝑒𝑖0 𝑒𝑖𝜙𝑘 𝑒𝑖2𝜙𝑘 𝑒𝑖(𝑁−1)𝜙𝑘 𝑒𝑖 𝑁−2 𝜙𝑘 𝑠𝑘 K sources (Input Vector) N receivers (Output vector) Steering Matrix (N x K) Multiple transmitters
- 36. Detecting AoA of K sources simultaneously
- 37. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 𝑒𝑖0 𝑒𝑖𝜙1 𝑒𝑖2𝜙1 𝑒𝑖(𝑁−1)𝜙1 𝑒𝑖 𝑁−2 𝜙1 𝑠1 = 𝑒𝑖0 𝑒𝑖𝜙2 𝑒𝑖2𝜙2 𝑒𝑖(𝑁−1)𝜙2 𝑒𝑖 𝑁−2 𝜙2 𝑠2 𝑒𝑖0 𝑒𝑖𝜙𝑘 𝑒𝑖2𝜙𝑘 𝑒𝑖(𝑁−1)𝜙𝑘 𝑒𝑖 𝑁−2 𝜙𝑘 𝑠𝑘
- 38. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 𝑒𝑖0 𝑒𝑖𝜙1 𝑒𝑖2𝜙1 𝑒𝑖(𝑁−1)𝜙1 𝑒𝑖 𝑁−2 𝜙1 𝑠1 = 𝑒𝑖0 𝑒𝑖𝜙2 𝑒𝑖2𝜙2 𝑒𝑖(𝑁−1)𝜙2 𝑒𝑖 𝑁−2 𝜙2 𝑠2 𝑒𝑖0 𝑒𝑖𝜙𝑘 𝑒𝑖2𝜙𝑘 𝑒𝑖(𝑁−1)𝜙𝑘 𝑒𝑖 𝑁−2 𝜙𝑘 𝑠𝑘 Multiply by conjugate of steering vector of source 1 𝑒−𝑖(𝑁−1)𝜙1 𝑒𝑖0 𝑒−𝑖𝜙1 𝑒−𝑖2𝜙1 .. 𝑒−𝑖(𝑁−1)𝜙1 𝑒𝑖0 𝑒−𝑖𝜙1 𝑒−𝑖2𝜙1 ..
- 39. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 𝑠1 = 𝑠2 𝑠𝑘 𝑁 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 𝑒−𝑖(𝑁−1)𝜙1 𝑒𝑖0 𝑒−𝑖𝜙1 𝑒−𝑖2𝜙1 ..
- 40. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 = 𝑠1 ∗ 𝑁 + 𝑠2 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + 𝑠3 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + … … . . All energy from direction 𝜃1(𝑓𝑟𝑜𝑚 𝑠1) have been aggregated and amplified A(𝜃1) = 𝑒−𝑖(𝑁−1)𝜙1 𝑒𝑖0 𝑒−𝑖𝜙1 𝑒−𝑖2𝜙1 ..
- 41. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 = 𝑠1 ∗ (𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒) + 𝑠2 ∗ 𝑁 + 𝑠3 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + … … . . All energy from direction 𝜃2(𝑓𝑟𝑜𝑚 𝑠2) have been aggregated and amplified A(𝜃2) = 𝑒−𝑖(𝑁−1)𝜙2 𝑒𝑖0 𝑒−𝑖𝜙2 𝑒−𝑖2𝜙2..
- 42. 𝑅𝑥1 𝑅𝑥2 𝑅𝑥3 𝑅𝑥𝑁 𝑅𝑥𝑁−1 = 𝑠1 ∗ (𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒) + 𝑠2 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + 𝑠3 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + … … . . The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals A(𝜃𝑟) = 𝑒−𝑖(𝑁−1)𝜙𝑟 𝑒𝑖0 𝑒−𝑖𝜙𝑟 𝑒−𝑖2𝜙𝑟..
- 43. • Construct a graph of for all values of • Any active source from direction should have a peak in the above graph .. • This is called delay and sum beamforming A(𝜃) 𝜃 𝜃𝑠
- 44. Detecting multiple AoA Suc A(𝜃) 𝑻𝒙𝟏 𝑻𝒙𝟐 𝑻𝒙𝟑 AoA Spectrum
- 45. Close by AoAs cannot be resolved 𝑻𝒙𝟏 𝑻𝒙𝟐 𝑻𝒙𝟑
- 46. MUSIC algorithm has sharp peaks to resolve close AoA Based on eigen decomposition and PCA – reference to be provided 𝐴𝑚𝑢𝑠𝑖𝑐(𝜃) 𝑻𝒙𝟏 𝑻𝒙𝟐 𝑻𝒙𝟑
- 47. Degrees of freedom for beamforming • Antenna separation • Initial phases of antenna sources • Number of antennas