Beamforming

Tx1

Tx1 cos⁡(2𝜋𝑓𝑡)

Tx2 Tx1 cos⁡(2𝜋𝑓𝑡) 𝝀 𝟐

𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝅 Tx2 Tx1 cos⁡(2𝜋𝑓𝑡) Rx 𝝀 𝟐

Destructive superimposition Tx2 Tx1 cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡+𝜋 =0 𝝀 𝟐 Zero signal

Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + Tx2 𝝀 𝟐 Tx1

Constructive superimposition cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡+0 =2cos⁡(2𝜋𝑓𝑡) Amplified signal (twice amplitude) Tx2 𝝀 𝟐 Tx1

Receiver at arbitrary location Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓 Tx2 𝝀 𝟐 Tx1

Arbitrary location, what’s the path difference

Path difference Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓 𝜃 Tx2 𝒅 Tx1

Path difference and phase difference Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓 𝜙(𝑝ℎ𝑎𝑠𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒)= 2𝜋 𝜆 ∗(𝑝𝑎𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒) Path difference = 𝒅𝒄𝒐𝒔(𝜽) 𝜙= 2𝜋 𝜆 ∗𝒅𝒄𝒐𝒔(𝜽) 𝜃 Tx2 𝒅 Tx1 𝐑𝐱 𝜽 =𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+ 𝟐𝝅 𝝀 𝒅𝒄𝒐𝒔(𝜽)

(𝑑= 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle

Radiation pattern: Rx amplitude as a function of angle (𝑑=𝜆)

Radiation pattern: Rx amplitude as a function of angle (𝑑=2𝜆)

Radiation pattern: Rx amplitude as a function of angle (𝑑= 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓

Radiation pattern: Rx amplitude as a function of angle (𝑑= 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle The initial phases can be controlled 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+ 𝝓 𝒊𝒏 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓

Radiation pattern: Rx amplitude as a function of angle (𝑑= 𝜆 2 ) Radiation pattern: Rx amplitude as a function of angle 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+ 𝝓 𝒊𝒏 + 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓 𝝓 𝒊𝒏 =0 𝝓 𝒊𝒏 =-x A non zero initial phase can change the radiation pattern 𝝓 𝒊𝒏 =0 𝝓 𝒊𝒏 =-x

Multiple antennas

. . . Rx 𝜃 𝒅 𝒅 Tx(N) Tx(N-1) Tx2 Tx1 2𝜋 𝑑𝑐𝑜𝑠(𝜃) 𝜆 𝑅𝑥= cos 2𝜋𝑓𝑡 …….. + cos 2𝜋𝑓𝑡+ 𝑁−2 ∗𝜙 + cos 2𝜋𝑓𝑡+ 𝑁−1 ∗𝜙

𝑅𝑥= cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡+𝜙 +cos⁡(2𝜋𝑓𝑡+2𝜙) + …… 𝑅𝑥= cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡+𝜙 +cos⁡(2𝜋𝑓𝑡+2𝜙) + …….. + cos 2𝜋𝑓𝑡+ 𝑁−2 ∗𝜙 + cos⁡(2𝜋𝑓𝑡+ 𝑁−1 ∗𝜙) cos 2𝜋𝑓𝑡 = 𝑒 𝑖2𝜋𝑓𝑡 + 𝑒 −𝑖2𝜋𝑓𝑡 2 =Re { e i2𝜋𝑓𝑡 } 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡+𝜙 + e i2𝜋𝑓𝑡+2𝜙 + …….. e i2𝜋𝑓𝑡+ 𝑁−1 𝜙 + e i2𝜋𝑓𝑡+ 𝑁−1 𝜙 } 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡 𝑒 𝑖𝜙 + e i2𝜋𝑓𝑡 𝑒 𝑖2𝜙 + …….. e i2𝜋𝑓𝑡 𝑒 𝑖 𝑁−2 𝜙 + e i2𝜋𝑓𝑡 𝑒 𝑖 𝑁−1 𝜙 } 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 1 + 𝑒 𝑖𝜙 + 𝑒 𝑖2𝜙 + …….. + 𝑒 𝑖 𝑁−2 𝜙 + 𝑒 𝑖 𝑁−1 𝜙 ) 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 1− 𝑒 𝑖𝑁𝜙 1 − 𝑒 𝑖𝜙 } 𝑹𝒙(𝜽)=𝑹𝒆{ 𝐞 𝐢𝟐𝝅𝒇𝒕 𝟏− 𝒆 𝒊𝑵 𝟐𝝅𝒅𝒄𝒐𝒔(𝜽) 𝝀 𝟏 − 𝒆 𝒊 𝟐𝝅𝒅𝒄𝒐𝒔(𝜽) 𝝀 }

(𝑑= 𝜆 2 ) Radiation pattern (𝑁=8) (𝑁=2) (𝑁=4)

Rotating the beam 𝑅𝑥= cos 2𝜋𝑓𝑡 + cos 2𝜋𝑓𝑡+𝜙 +cos⁡(2𝜋𝑓𝑡+2𝜙) + …….. + cos 2𝜋𝑓𝑡+ 𝑁−2 ∗𝜙 + cos⁡(2𝜋𝑓𝑡+ 𝑁−1 ∗𝜙) 𝑅𝑥= cos 2𝜋𝑓𝑡+ 𝜙 𝑖𝑛𝑜 + cos 2𝜋𝑓𝑡+𝜙+ 𝜙 𝑖𝑛1 +cos⁡(2𝜋𝑓𝑡+2𝜙+ 𝜙 𝑖𝑛2 ) + … + cos 2𝜋𝑓𝑡+ 𝑁−2 ∗𝜙+ 𝜙 𝑖𝑛(𝑁−2) + cos⁡(2𝜋𝑓𝑡+ 𝑁−1 ∗𝜙+ 𝜙 𝑖𝑛(𝑁−1) ) 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡+𝜙+ 𝜙 𝑖𝑛0 + e i2𝜋𝑓𝑡+2𝜙+ 2𝜙 𝑖𝑛1 + …….. e i2𝜋𝑓𝑡+ 𝑁−2 𝜙+ 𝜙 𝑖𝑛(𝑁−2) + e i2𝜋𝑓𝑡+ 𝑁−1 𝜙+ 𝜙 𝑖𝑛(𝑁−1 } 𝜙 𝑖𝑛𝑜 =0, 𝜙 𝑖𝑛1 = 𝜙 𝑖𝑛 , 𝜙 𝑖𝑛2 = 2𝜙 𝑖𝑛 ……….., 𝜙 𝑖𝑛1 = (𝑁−1)∗𝜙 𝑖𝑛 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡+𝜙+ 𝜙 𝑖𝑛 + e i2𝜋𝑓𝑡+2𝜙+ 2𝜙 𝑖𝑛 + …….. e i2𝜋𝑓𝑡+ 𝑁−2 𝜙+ (𝑁−2)𝜙 𝑖𝑛 + e i2𝜋𝑓𝑡+ 𝑁−1 𝜙+ (𝑁−2)𝜙 𝑖𝑛 } Goal is to move the maxima to a different angle theta .. 𝜙= 2𝜋 𝜆 ∗𝒅𝒄𝒐𝒔(𝜽) 𝑆𝑒𝑡 𝜙 𝑖𝑛 =−𝜙=− 2𝜋 𝜆 ∗𝒅𝒄𝒐𝒔(𝜽) 𝑅𝑥=𝑅𝑒{ e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡 + …….. e i2𝜋𝑓𝑡 + e i2𝜋𝑓𝑡 } 𝑅𝑥=𝑅𝑒{ Ne i2𝜋𝑓𝑡 } A maxima occurs in the direction of 𝜽

Rotating the beam 𝝓 𝒊𝒏 =−𝝓=− 𝟐𝝅 𝝀 ∗𝒅𝒄𝒐𝒔(𝟔𝟎) 𝝓 𝒊𝒏 =−𝝓=− 𝟐𝝅 𝝀 ∗𝒅𝒄𝒐𝒔(𝟒𝟓)

Networking applications

Acoustic Beamforming – noise suppression Silent zone Audible Zone

Other applications Localization Gesture tracking RF Imaging

Reception

Sensing Angle of Arrival (AoA) Tx Path difference = 𝒅𝒄𝒐𝒔(𝜽) 𝜃 Rx1 𝒅 Rx2 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕+𝝓 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕 𝜙= 2𝜋 𝜆 ∗𝒅𝒄𝒐𝒔(𝜽) 𝜽(𝑨𝒐𝑨)=𝒂𝒄𝒐𝒔 𝝀𝝓 𝟐𝝅𝒅

Antenna array . . . Tx 𝜃 𝒅 𝒅 Rx(N) Rx(N-1) Rx2 Rx1 cos 2𝜋𝑓𝑡 cos 2𝜋𝑓𝑡+𝜙 2𝜋 𝑑𝑐𝑜𝑠(𝜃) 𝜆

𝑒 𝑖2𝜋𝑓𝑡 𝑒 𝑖0 cos 2𝜋𝑓𝑡 𝑒 𝑖0 𝑅 𝑥 1 𝑒 𝑖2𝜋𝑓𝑡+𝜙 𝑒 𝑖𝜙 𝑅 𝑥 2 𝑒 𝑖𝜙 cos 2𝜋𝑓𝑡+𝜙 𝑒 𝑖2𝜋𝑓𝑡+2𝜙 𝑒 𝑖2𝜙 𝑅 𝑥 3 𝑒 𝑖2𝜙 cos 2𝜋𝑓𝑡+2𝜙 𝑒 𝑖2𝜋𝑓𝑡 𝑠 𝑡 = = = = cos 2𝜋𝑓𝑡+(𝑁−2)𝜙 𝑒 𝑖(𝑁−2)𝜙 𝑅 𝑥 𝑁−1 𝑒 𝑖2𝜋𝑓𝑡+(𝑁−2)𝜙 𝑒 𝑖(𝑁−2)𝜙 𝑅 𝑥 𝑁 cos 2𝜋𝑓𝑡+(𝑁−1)𝜙 𝑒 𝑖𝜙 𝑒 𝑖2𝜋𝑓𝑡+(𝑁−1)𝜙 𝑒 𝑖𝜙

2𝜋 𝑑𝑐𝑜𝑠(𝜃) 𝜆 𝑒 𝑖0 𝑅 𝑥 1 𝑅 𝑥 2 𝑒 𝑖𝜙 𝑅 𝑥 3 𝑒 𝑖2𝜙 𝑠 𝑡 = 𝑅 𝑥 𝑁−1 𝑒 𝑖(𝑁−2)𝜙 𝑅 𝑥 𝑁 𝑒 𝑖𝜙 Steering vector

Multiple transmitters Tx2 Tx1 . . . 𝜃 𝒅 𝒅 Rx(N) Rx(N-1) Rx2 Rx1

Multiple transmitters 2𝜋 𝑑𝑐𝑜𝑠( 𝜃 1 ) 𝜆 2𝜋 𝑑𝑐𝑜𝑠( 𝜃 2 ) 𝜆 2𝜋 𝑑𝑐𝑜𝑠( 𝜃 𝑘 ) 𝜆 𝑅 𝑥 1 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑅 𝑥 2 𝑒 𝑖 𝜙 1 𝑒 𝑖 𝜙 2 𝑒 𝑖 𝜙 𝑘 𝑒 𝑖2 𝜙 𝑘 𝑅 𝑥 3 𝑒 𝑖2 𝜙 1 𝑒 𝑖2 𝜙 2 𝑠 1 𝑠 2 𝑠 𝑘 = + + 𝑅 𝑥 𝑁−1 𝑒 𝑖 𝑁−2 𝜙 1 𝑒 𝑖 𝑁−2 𝜙 2 𝑒 𝑖 𝑁−2 𝜙 𝑘 𝑅 𝑥 𝑁 𝑒 𝑖 (𝑁−1)𝜙 1 𝑒 𝑖 (𝑁−1)𝜙 2 𝑒 𝑖 (𝑁−1)𝜙 𝑘 Output is a linear combination of steering vectors from different directions

Multiple transmitters 𝑅 𝑥 1 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑠 1 𝑅 𝑥 2 𝑒 𝑖 𝜙 1 𝑒 𝑖 𝜙 2 𝑒 𝑖 𝜙 𝑘 𝑠 2 𝑅 𝑥 3 𝑒 𝑖2 𝜙 1 𝑒 𝑖2 𝜙 2 𝑒 𝑖2 𝜙 𝑘 = 𝑅 𝑥 𝑁−1 𝑒 𝑖 𝑁−2 𝜙 1 𝑒 𝑖 𝑁−2 𝜙 2 𝑒 𝑖 𝑁−2 𝜙 𝑘 𝑅 𝑥 𝑁 𝑒 𝑖 (𝑁−1)𝜙 1 𝑒 𝑖 (𝑁−1)𝜙 2 𝑒 𝑖 (𝑁−1)𝜙 𝑘 𝑠 𝑘 Steering Matrix (N x K) K sources (Input Vector) N receivers (Output vector)

Detecting AoA of K sources simultaneously

𝑅 𝑥 1 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑠 1 𝑅 𝑥 2 𝑒 𝑖 𝜙 1 𝑒 𝑖 𝜙 2 𝑒 𝑖 𝜙 𝑘 𝑠 2 𝑅 𝑥 3 𝑒 𝑖2 𝜙 1 𝑒 𝑖2 𝜙 2 𝑒 𝑖2 𝜙 𝑘 = 𝑅 𝑥 𝑁−1 𝑒 𝑖 𝑁−2 𝜙 1 𝑒 𝑖 𝑁−2 𝜙 2 𝑒 𝑖 𝑁−2 𝜙 𝑘 𝑅 𝑥 𝑁 𝑒 𝑖 (𝑁−1)𝜙 1 𝑒 𝑖 (𝑁−1)𝜙 2 𝑒 𝑖 (𝑁−1)𝜙 𝑘 𝑠 𝑘

Multiply by conjugate of steering vector of source 1 𝑅 𝑥 1 𝑒 −𝑖 (𝑁−1)𝜙 1 𝑒 𝑖0 𝑒 −𝑖 𝜙 1 𝑒 −𝑖 2𝜙 1 .. 𝑒 −𝑖 (𝑁−1)𝜙 1 𝑒 𝑖0 𝑒 −𝑖 𝜙 1 𝑒 −𝑖 2𝜙 1 .. 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑠 1 𝑅 𝑥 2 𝑒 𝑖 𝜙 1 𝑒 𝑖 𝜙 2 𝑒 𝑖 𝜙 𝑘 𝑠 2 𝑅 𝑥 3 𝑒 𝑖2 𝜙 1 𝑒 𝑖2 𝜙 2 𝑒 𝑖2 𝜙 𝑘 = 𝑅 𝑥 𝑁−1 𝑒 𝑖 𝑁−2 𝜙 1 𝑒 𝑖 𝑁−2 𝜙 2 𝑒 𝑖 𝑁−2 𝜙 𝑘 𝑅 𝑥 𝑁 𝑒 𝑖 (𝑁−1)𝜙 1 𝑒 𝑖 (𝑁−1)𝜙 2 𝑒 𝑖 (𝑁−1)𝜙 𝑘 𝑠 𝑘

𝑒 −𝑖 (𝑁−1)𝜙 1 𝑒 𝑖0 𝑒 −𝑖 𝜙 1 𝑒 −𝑖 2𝜙 1 .. 𝑅 𝑥 1 𝑠 1 𝑁 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 𝑅 𝑥 2 𝑠 2 𝑅 𝑥 3 = 𝑅 𝑥 𝑁−1 𝑅 𝑥 𝑁 𝑠 𝑘

𝑅 𝑥 1 𝑒 −𝑖 (𝑁−1)𝜙 1 𝑒 𝑖0 𝑒 −𝑖 𝜙 1 𝑒 −𝑖 2𝜙 1 .. A( 𝜃 1 ) = 𝑅 𝑥 2 𝑅 𝑥 3 = 𝑠 1 ∗𝑁+ 𝑠 2 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + 𝑠 3 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + …….. 𝑅 𝑥 𝑁−1 𝑅 𝑥 𝑁 All energy from direction 𝜃 1 ( 𝑓𝑟𝑜𝑚 𝑠 1 ) have been aggregated and amplified

𝑅 𝑥 1 𝑒 −𝑖 (𝑁−1)𝜙 2 𝑒 𝑖0 𝑒 −𝑖 𝜙 2 𝑒 −𝑖 2𝜙 2 .. A( 𝜃 2 ) = 𝑅 𝑥 2 𝑅 𝑥 3 = 𝑠 1 ∗(𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒)+ 𝑠 2 ∗ 𝑁 + 𝑠 3 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + …….. 𝑅 𝑥 𝑁−1 𝑅 𝑥 𝑁 All energy from direction 𝜃 2 ( 𝑓𝑟𝑜𝑚 𝑠 2 ) have been aggregated and amplified

𝑅 𝑥 1 𝑒 −𝑖 (𝑁−1)𝜙 𝑟 𝑒 𝑖0 𝑒 −𝑖 𝜙 𝑟 𝑒 −𝑖 2𝜙 𝑟 .. A( 𝜃 𝑟 ) = 𝑅 𝑥 2 𝑅 𝑥 3 = 𝑠 1 ∗(𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒)+ 𝑠 2 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + 𝑠 3 ∗ 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 + …….. 𝑅 𝑥 𝑁−1 𝑅 𝑥 𝑁 The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals

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 𝜃 𝑠

Detecting multiple AoA 𝑻 𝒙 𝟐 𝑻 𝒙 𝟑 A(𝜃) AoA Spectrum 𝑻 𝒙 𝟏 Suc

Close by AoAs cannot be resolved 𝑻 𝒙 𝟑 𝑻 𝒙 𝟐 𝑻 𝒙 𝟏

MUSIC algorithm has sharp peaks to resolve close AoA Based on eigen decomposition and PCA – reference to be provided 𝐴 𝑚𝑢𝑠𝑖𝑐 (𝜃) 𝑻 𝒙 𝟑 𝑻 𝒙 𝟐 𝑻 𝒙 𝟏

Degrees of freedom for beamforming Antenna separation Initial phases of antenna sources Number of antennas