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Beamforming
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Tx1
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Tx1 cosβ‘(2πππ‘)
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Tx2 Tx1 cosβ‘(2πππ‘) π π
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ππ¨π¬ ππ
ππ + ππ¨π¬ ππ
ππ+π
Tx2 Tx1 cosβ‘(2πππ‘) Rx π π
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Destructive superimposition
Tx2 Tx1 cos 2πππ‘ + cos 2πππ‘+π =0 π π Zero signal
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Rx ππ¨π¬ ππ
ππ + Tx2 π π Tx1
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Constructive superimposition
cos 2πππ‘ + cos 2πππ‘+0 =2cosβ‘(2πππ‘) Amplified signal (twice amplitude) Tx2 π π Tx1
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Receiver at arbitrary location
Rx ππ¨π¬ ππ
ππ + ππ¨π¬ ππ
ππ+π Tx2 π π Tx1
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Arbitrary location, whatβs the path difference
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Path difference Rx ππ¨π¬ ππ
ππ + ππ¨π¬ ππ
ππ+π π Tx2 π
Tx1
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Path difference and phase difference
Rx ππ¨π¬ ππ
ππ + ππ¨π¬ ππ
ππ+π π(πβππ π ππππππππππ)= 2π π β(πππ‘β ππππππππππ) Path difference = π
πππ(π½) π= 2π π βπ
πππ(π½) π Tx2 π
Tx1 ππ± π½ =ππ¨π¬ ππ
ππ + ππ¨π¬ ππ
ππ+ ππ
π π
πππ(π½)
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(π= π 2 ) Radiation pattern: Rx amplitude as a function of angle
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Radiation pattern: Rx amplitude as a function of angle
(π=π)
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Radiation pattern: Rx amplitude as a function of angle
(π=2π)
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Radiation pattern: Rx amplitude as a function of angle
(π= π 2 ) Radiation pattern: Rx amplitude as a function of angle ππ¨π¬ ππ
ππ + ππ¨π¬ ππ
ππ+π
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Radiation pattern: Rx amplitude as a function of angle
(π= π 2 ) Radiation pattern: Rx amplitude as a function of angle The initial phases can be controlled ππ¨π¬ ππ
ππ+ π ππ + ππ¨π¬ ππ
ππ+π
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Radiation pattern: Rx amplitude as a function of angle
(π= π 2 ) Radiation pattern: Rx amplitude as a function of angle ππ¨π¬ ππ
ππ+ π ππ + ππ¨π¬ ππ
ππ+π π ππ =0 π ππ =-x A non zero initial phase can change the radiation pattern π ππ =0 π ππ =-x
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Multiple antennas
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. . . Rx π π
π
Tx(N) Tx(N-1) Tx2 Tx1 2π ππππ (π) π π
π₯= cos 2πππ‘
β¦β¦.. + cos 2πππ‘+ πβ2 βπ + cos 2πππ‘+ πβ1 βπ
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π
π₯= cos 2πππ‘ + cos 2πππ‘+π +cosβ‘(2πππ‘+2π) + β¦β¦
π
π₯= cos 2πππ‘ + cos 2πππ‘+π +cosβ‘(2πππ‘+2π) + β¦β¦.. + cos 2πππ‘+ πβ2 βπ + cosβ‘(2πππ‘+ πβ1 βπ) cos 2πππ‘ = π π2πππ‘ + π βπ2πππ‘ 2 =Re { e i2πππ‘ } π
π₯=π
π{ e i2πππ‘ + e i2πππ‘+π + e i2πππ‘+2π + β¦β¦.. e i2πππ‘+ πβ1 π + e i2πππ‘+ πβ1 π } π
π₯=π
π{ e i2πππ‘ + e i2πππ‘ π ππ + e i2πππ‘ π π2π + β¦β¦.. e i2πππ‘ π π πβ2 π + e i2πππ‘ π π πβ1 π } π
π₯=π
π{ e i2πππ‘ 1 + π ππ + π π2π + β¦β¦.. + π π πβ2 π + π π πβ1 π ) π
π₯=π
π{ e i2πππ‘ 1β π πππ 1 β π ππ } πΉπ(π½)=πΉπ{ π π’ππ
ππ πβ π ππ΅ ππ
π
πππ(π½) π π β π π ππ
π
πππ(π½) π }
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(π= π 2 ) Radiation pattern (π=8) (π=2) (π=4)
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Rotating the beam π
π₯= cos 2πππ‘ + cos 2πππ‘+π +cosβ‘(2πππ‘+2π) + β¦β¦.. + cos 2πππ‘+ πβ2 βπ + cosβ‘(2πππ‘+ πβ1 βπ) π
π₯= cos 2πππ‘+ π πππ + cos 2πππ‘+π+ π ππ1 +cosβ‘(2πππ‘+2π+ π ππ2 ) + β¦ + cos 2πππ‘+ πβ2 βπ+ π ππ(πβ2) + cosβ‘(2πππ‘+ πβ1 βπ+ π ππ(πβ1) ) π
π₯=π
π{ e i2πππ‘ + e i2πππ‘+π+ π ππ0 + e i2πππ‘+2π+ 2π ππ1 + β¦β¦.. e i2πππ‘+ πβ2 π+ π ππ(πβ2) + e i2πππ‘+ πβ1 π+ π ππ(πβ1 } π πππ =0, π ππ1 = π ππ , π ππ2 = 2π ππ β¦β¦β¦.., π ππ1 = (πβ1)βπ ππ π
π₯=π
π{ e i2πππ‘ + e i2πππ‘+π+ π ππ + e i2πππ‘+2π+ 2π ππ + β¦β¦.. e i2πππ‘+ πβ2 π+ (πβ2)π ππ + e i2πππ‘+ πβ1 π+ (πβ2)π ππ } Goal is to move the maxima to a different angle theta .. π= 2π π βπ
πππ(π½) πππ‘ π ππ =βπ=β 2π π βπ
πππ(π½) π
π₯=π
π{ e i2πππ‘ + e i2πππ‘ + e i2πππ‘ + β¦β¦.. e i2πππ‘ + e i2πππ‘ } π
π₯=π
π{ Ne i2πππ‘ } A maxima occurs in the direction of π½
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Rotating the beam π ππ =βπ=β ππ
π βπ
πππ(ππ) π ππ =βπ=β ππ
π βπ
πππ(ππ)
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Networking applications
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Acoustic Beamforming β noise suppression
Silent zone Audible Zone
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Other applications Localization Gesture tracking RF Imaging
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Reception
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Sensing Angle of Arrival (AoA)
Tx Path difference = π
πππ(π½) π Rx1 π
Rx2 ππ¨π¬ ππ
ππ+π ππ¨π¬ ππ
ππ π= 2π π βπ
πππ(π½) π½(π¨ππ¨)=ππππ ππ ππ
π
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Antenna array . . . Tx π π
π
Rx(N) Rx(N-1) Rx2 Rx1 cos 2πππ‘ cos 2πππ‘+π
2π ππππ (π) π
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π π2πππ‘ π π0 cos 2πππ‘ π π0 π
π₯ 1 π π2πππ‘+π π ππ π
π₯ 2 π ππ cos 2πππ‘+π π π2πππ‘+2π π π2π π
π₯ 3 π π2π cos 2πππ‘+2π π π2πππ‘ π π‘ = = = = cos 2πππ‘+(πβ2)π π π(πβ2)π π
π₯ πβ1 π π2πππ‘+(πβ2)π π π(πβ2)π π
π₯ π cos 2πππ‘+(πβ1)π π ππ π π2πππ‘+(πβ1)π π ππ
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2π ππππ (π) π π π0 π
π₯ 1 π
π₯ 2 π ππ π
π₯ 3 π π2π π π‘ = π
π₯ πβ1 π π(πβ2)π π
π₯ π π ππ Steering vector
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Multiple transmitters
Tx2 Tx1 . . . π π
π
Rx(N) Rx(N-1) Rx2 Rx1
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Multiple transmitters
2π ππππ ( π 1 ) π 2π ππππ ( π 2 ) π 2π ππππ ( π π ) π π
π₯ 1 π π0 π π0 π π0 π
π₯ 2 π π π 1 π π π 2 π π π π π π2 π π π
π₯ 3 π π2 π 1 π π2 π 2 π 1 π 2 π π = + + π
π₯ πβ1 π π πβ2 π 1 π π πβ2 π 2 π π πβ2 π π π
π₯ π π π (πβ1)π 1 π π (πβ1)π 2 π π (πβ1)π π Output is a linear combination of steering vectors from different directions
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Multiple transmitters
π
π₯ 1 π π0 π π0 π π0 π 1 π
π₯ 2 π π π 1 π π π 2 π π π π π 2 π
π₯ 3 π π2 π 1 π π2 π 2 π π2 π π = π
π₯ πβ1 π π πβ2 π 1 π π πβ2 π 2 π π πβ2 π π π
π₯ π π π (πβ1)π 1 π π (πβ1)π 2 π π (πβ1)π π π π Steering Matrix (N x K) K sources (Input Vector) N receivers (Output vector)
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Detecting AoA of K sources simultaneously
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π
π₯ 1 π π0 π π0 π π0 π 1 π
π₯ 2 π π π 1 π π π 2 π π π π π 2 π
π₯ 3 π π2 π 1 π π2 π 2 π π2 π π = π
π₯ πβ1 π π πβ2 π 1 π π πβ2 π 2 π π πβ2 π π π
π₯ π π π (πβ1)π 1 π π (πβ1)π 2 π π (πβ1)π π π π
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Multiply by conjugate of steering vector of source 1
π
π₯ 1 π βπ (πβ1)π 1 π π0 π βπ π 1 π βπ 2π 1 .. π βπ (πβ1)π 1 π π0 π βπ π 1 π βπ 2π 1 .. π π0 π π0 π π0 π 1 π
π₯ 2 π π π 1 π π π 2 π π π π π 2 π
π₯ 3 π π2 π 1 π π2 π 2 π π2 π π = π
π₯ πβ1 π π πβ2 π 1 π π πβ2 π 2 π π πβ2 π π π
π₯ π π π (πβ1)π 1 π π (πβ1)π 2 π π (πβ1)π π π π
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π βπ (πβ1)π 1 π π0 π βπ π 1 π βπ 2π 1 .. π
π₯ 1 π 1 π π ππππ π£πππ’π π ππππ π£πππ’π π
π₯ 2 π 2 π
π₯ 3 = π
π₯ πβ1 π
π₯ π π π
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π
π₯ 1 π βπ (πβ1)π 1 π π0 π βπ π 1 π βπ 2π 1 .. A( π 1 ) = π
π₯ 2 π
π₯ 3 = π 1 βπ+ π 2 β π ππππ π£πππ’π + π 3 β π ππππ π£πππ’π + β¦β¦.. π
π₯ πβ1 π
π₯ π All energy from direction π 1 ( ππππ π 1 ) have been aggregated and amplified
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π
π₯ 1 π βπ (πβ1)π 2 π π0 π βπ π 2 π βπ 2π 2 .. A( π 2 ) = π
π₯ 2 π
π₯ 3 = π 1 β(π ππππ π£πππ’π)+ π 2 β π + π 3 β π ππππ π£πππ’π + β¦β¦.. π
π₯ πβ1 π
π₯ π All energy from direction π 2 ( ππππ π 2 ) have been aggregated and amplified
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π
π₯ 1 π βπ (πβ1)π π π π0 π βπ π π π βπ 2π π .. A( π π ) = π
π₯ 2 π
π₯ 3 = 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¦.. π
π₯ πβ1 π
π₯ π The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals
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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(π) π π π
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Detecting multiple AoA
π» π π π» π π A(π) AoA Spectrum π» π π Suc
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Close by AoAs cannot be resolved
π» π π π» π π π» π π
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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» π π π» π π π» π π
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Degrees of freedom for beamforming
Antenna separation Initial phases of antenna sources Number of antennas
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