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beamforming.pptx
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beamforming.pptx
- Beamforming
- Tx1
- Tx1 cos(2πππ‘)
- Tx1 Tx2 π π cos(2πππ‘)
- Tx1 Tx2 π π cos(2πππ‘) Rx ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ + π
- Destructive superimposition Tx1 Tx2 π π Zero signal cos 2πππ‘ + cos 2πππ‘ + π = 0
- Tx1 Tx2 π π Rx ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ
- Constructive superimposition Tx1 Tx2 π π Amplified signal (twice amplitude) cos 2πππ‘ + cos 2πππ‘ + 0 = 2cos(2πππ‘)
- Receiver at arbitrary location Tx1 Tx2 π π ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ + π Rx
- Arbitrary location, whatβs the path difference Path difference = ??
- Path difference Tx1 Tx2 π π ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ + π Rx
- Path difference and phase difference Tx1 Tx2 π Path difference = π π πππ(π½) π(πβππ π ππππππππππ) = 2π π β (πππ‘β ππππππππππ) π = 2π π β π πππ(π½) ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ + π Rx ππ± π½ = ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ + ππ π π πππ(π½)
- (π = π 2 ) Radiation pattern: Rx amplitude as a function of angle
- (π = π) Radiation pattern: Rx amplitude as a function of angle
- Radiation pattern: Rx amplitude as a function of angle (π = 2π)
- (π = π 2 ) Radiation pattern: Rx amplitude as a function of angle ππ¨π¬ ππ ππ + ππ¨π¬ ππ ππ + π
- (π = π 2 ) Radiation pattern: Rx amplitude as a function of angle ππ¨π¬ ππ ππ + πππ + ππ¨π¬ ππ ππ + π The initial phases can be controlled
- Radiation pattern: Rx amplitude as a function of angle (π = π 2 ) πππ=0 πππ=-x ππ¨π¬ ππ ππ + πππ + ππ¨π¬ ππ ππ + π πππ=0 πππ=-x A non zero initial phase can change the radiation pattern
- Multiple antennas
- Tx1 Tx2 π π Tx(N-1) π . . . 2π ππππ (π) π Tx(N) Rx cos 2πππ‘ + cos 2πππ‘ + π + cos 2πππ‘ + 2π + cos 2πππ‘ + π β 2 β π + cos 2πππ‘ + π β 1 β π π π₯ = β¦β¦..
- π π₯ = cos 2πππ‘ + cos 2πππ‘ + π + cos(2πππ‘ + 2π) + β¦ β¦ . . + cos 2πππ‘ + π β 2 β π + cos(2πππ‘ + π β 1 β π) cos 2πππ‘ = ππ2πππ‘ + πβπ2πππ‘ 2 = Re {ei2πππ‘} π π₯ = π π{ei2πππ‘ + ei2πππ‘+π + ei2πππ‘+2π + β¦ β¦ . . ei2πππ‘+ πβ1 π + ei2πππ‘+ πβ1 π} π π₯ = π π{ei2πππ‘ + ei2πππ‘πππ + ei2πππ‘ππ2π + β¦ β¦ . . ei2πππ‘ππ πβ2 π + ei2πππ‘ππ πβ1 π} π π₯ = π π{ei2πππ‘ 1 + πππ + ππ2π + β¦ β¦ . . + ππ πβ2 π + ππ πβ1 π ) π π₯ = π π{ei2πππ‘ 1 β ππππ 1 β πππ } πΉπ(π½) = πΉπ{ππ’ππ ππ π β π ππ΅ ππ π πππ(π½) π π β π π ππ π πππ(π½) π }
- Radiation pattern (π = π 2 ) (π = 2) (π = 4) (π = 8)
- π π₯ = cos 2πππ‘ + cos 2πππ‘ + π + cos(2πππ‘ + 2π) + β¦ β¦ . . + cos 2πππ‘ + π β 2 β π + cos(2πππ‘ + π β 1 β π) π π₯ = π π{ei2πππ‘ + ei2πππ‘+π+πππ + ei2πππ‘+2π+2πππ + β¦ β¦ . . ei2πππ‘+ πβ2 π+(πβ2)πππ + ei2πππ‘+ πβ1 π+(πβ2)πππ} π π₯ = cos 2πππ‘ + ππππ + cos 2πππ‘ + π + πππ1 + cos(2πππ‘ + 2π + πππ2) + β¦ + cos 2πππ‘ + π β 2 β π + πππ(πβ2) + cos(2πππ‘ + π β 1 β π + πππ(πβ1)) ππππ = 0, πππ1 = πππ , πππ2 = 2πππ β¦ β¦ β¦ . . , πππ1 = (π β 1) β πππ π = 2π π β π πππ(π½) πππ‘ πππ = βπ = β 2π π β π πππ(π½) π π₯ = π π{ei2πππ‘ + ei2πππ‘ + ei2πππ‘ + β¦ β¦ . . ei2πππ‘ + ei2πππ‘} π π₯ = π π{Nei2πππ‘} A maxima occurs in the direction of π½ Rotating the beam π π₯ = π π{ei2πππ‘ + ei2πππ‘+π+πππ0 + ei2πππ‘+2π+2πππ1 + β¦ β¦ . . ei2πππ‘+ πβ2 π+πππ(πβ2) + ei2πππ‘+ πβ1 π+πππ(πβ1}
- πππ = βπ = β ππ π β π πππ(ππ) πππ = βπ = β ππ π β π πππ(ππ) Rotating the beam
- Networking applications 25
- Acoustic Beamforming β noise suppression Silent zone Audible Zone
- Other applications β’ Localization β’ Gesture tracking β’ RF Imaging
- Reception
- Sensing Angle of Arrival (AoA) Rx2 Rx1 π Path difference = π π πππ(π½) π = 2π π β π πππ(π½) ππ¨π¬ ππ ππ + π Tx ππ¨π¬ ππ ππ π½(π¨ππ¨) = ππππ ππ ππ π
- Rx1 Rx2 π π Rx(N-1) π . . . 2π ππππ (π) π Rx(N) Tx cos 2πππ‘ cos 2πππ‘ + π cos(2πππ‘ + π β 1 β π) Antenna array
- cos 2πππ‘ cos 2πππ‘ + π cos 2πππ‘ + 2π cos 2πππ‘ + (π β 1)π cos 2πππ‘ + (π β 2)π π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 ππ2πππ‘ ππ2πππ‘+π ππ2πππ‘+2π ππ2πππ‘+(πβ1)π ππ2πππ‘+(πβ2)π ππ0 πππ ππ2π πππ ππ(πβ2)π ππ2πππ‘ = = = ππ0 πππ ππ2π πππ ππ(πβ2)π π π‘ =
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 = ππ0 πππ ππ2π πππ ππ(πβ2)π Steering vector π π‘ 2π ππππ (π) π
- Rx1 Rx2 π π Rx(N-1) π . . . Rx(N) Tx1 Tx2 Multiple transmitters
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 ππ0 πππ1 ππ2π1 ππ(πβ1)π1 ππ πβ2 π1 π 1 = ππ0 πππ2 ππ2π2 ππ(πβ1)π2 ππ πβ2 π2 π 2 + ππ0 ππππ ππ2ππ ππ(πβ1)ππ ππ πβ2 ππ π π + 2π ππππ (π1) π 2π ππππ (π2) π 2π ππππ (ππ) π Multiple transmitters Output is a linear combination of steering vectors from different directions
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 ππ0 πππ1 ππ2π1 ππ(πβ1)π1 ππ πβ2 π1 π 1 = ππ0 πππ2 ππ2π2 ππ(πβ1)π2 ππ πβ2 π2 π 2 ππ0 ππππ ππ2ππ ππ(πβ1)ππ ππ πβ2 ππ π π K sources (Input Vector) N receivers (Output vector) Steering Matrix (N x K) Multiple transmitters
- Detecting AoA of K sources simultaneously
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 ππ0 πππ1 ππ2π1 ππ(πβ1)π1 ππ πβ2 π1 π 1 = ππ0 πππ2 ππ2π2 ππ(πβ1)π2 ππ πβ2 π2 π 2 ππ0 ππππ ππ2ππ ππ(πβ1)ππ ππ πβ2 ππ π π
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 ππ0 πππ1 ππ2π1 ππ(πβ1)π1 ππ πβ2 π1 π 1 = ππ0 πππ2 ππ2π2 ππ(πβ1)π2 ππ πβ2 π2 π 2 ππ0 ππππ ππ2ππ ππ(πβ1)ππ ππ πβ2 ππ π π Multiply by conjugate of steering vector of source 1 πβπ(πβ1)π1 ππ0 πβππ1 πβπ2π1 .. πβπ(πβ1)π1 ππ0 πβππ1 πβπ2π1 ..
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 π 1 = π 2 π π π π ππππ π£πππ’π π ππππ π£πππ’π πβπ(πβ1)π1 ππ0 πβππ1 πβπ2π1 ..
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 = π 1 β π + π 2 β π ππππ π£πππ’π + π 3 β π ππππ π£πππ’π + β¦ β¦ . . All energy from direction π1(ππππ π 1) have been aggregated and amplified A(π1) = πβπ(πβ1)π1 ππ0 πβππ1 πβπ2π1 ..
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 = π 1 β (π ππππ π£πππ’π) + π 2 β π + π 3 β π ππππ π£πππ’π + β¦ β¦ . . All energy from direction π2(ππππ π 2) have been aggregated and amplified A(π2) = πβπ(πβ1)π2 ππ0 πβππ2 πβπ2π2..
- π π₯1 π π₯2 π π₯3 π π₯π π π₯πβ1 = π 1 β (π ππππ π£πππ’π) + π 2 β π ππππ π£πππ’π + π 3 β π ππππ π£πππ’π + β¦ β¦ . . The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals A(ππ) = πβπ(πβ1)ππ ππ0 πβπππ πβπ2ππ..
- β’ Construct a graph of for all values of β’ Any active source from direction should have a peak in the above graph .. β’ This is called delay and sum beamforming A(π) π ππ
- Detecting multiple AoA Suc A(π) π»ππ π»ππ π»ππ AoA Spectrum
- Close by AoAs cannot be resolved π»ππ π»ππ π»ππ
- MUSIC algorithm has sharp peaks to resolve close AoA Based on eigen decomposition and PCA β reference to be provided π΄ππ’π ππ(π) π»ππ π»ππ π»ππ
- Degrees of freedom for beamforming β’ Antenna separation β’ Initial phases of antenna sources β’ Number of antennas
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