Beamforming

Tx1

Tx1 cos(2𝜋𝑓𝑡)

Tx1
Tx2
𝝀
𝟐
cos(2𝜋𝑓𝑡)

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

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

Tx1
Tx2 𝝀
𝟐
Rx 𝐜𝐨𝐬 𝟐𝝅𝒇𝒕
+
𝐜𝐨𝐬 𝟐𝝅𝒇𝒕

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

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

Arbitrary location, what’s the path difference
Path difference = ??

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

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)

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

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

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

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

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

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

Multiple antennas

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 = ……..

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
}

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

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}

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

Networking applications 25

Acoustic Beamforming – noise suppression
Silent zone Audible Zone

Other applications
• Localization
• Gesture tracking
• RF Imaging

Reception

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

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

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
=

𝑅𝑥1
𝑅𝑥2
𝑅𝑥3
𝑅𝑥𝑁
𝑅𝑥𝑁−1
=
𝑒𝑖0
𝑒𝑖𝜙
𝑒𝑖2𝜙
𝑒𝑖𝜙
𝑒𝑖(𝑁−2)𝜙
Steering vector
𝑠𝑡
2𝜋
𝑑𝑐𝑜𝑠(𝜃)
𝜆

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

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

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

Detecting AoA of K sources simultaneously

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

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 ..

𝑅𝑥1
𝑅𝑥2
𝑅𝑥3
𝑅𝑥𝑁
𝑅𝑥𝑁−1
𝑠1
=
𝑠2
𝑠𝑘
𝑁 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠𝑚𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒
𝑒−𝑖(𝑁−1)𝜙1
𝑒𝑖0 𝑒−𝑖𝜙1 𝑒−𝑖2𝜙1 ..

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 ..

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..

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..

• 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

Detecting multiple AoA
Suc
A(𝜃)
𝑻𝒙𝟏
𝑻𝒙𝟐
𝑻𝒙𝟑
AoA Spectrum

Close by AoAs cannot be resolved
𝑻𝒙𝟏
𝑻𝒙𝟐
𝑻𝒙𝟑

MUSIC algorithm has sharp peaks to resolve close AoA
Based on eigen decomposition and PCA – reference to be provided
𝐴𝑚𝑢𝑠𝑖𝑐(𝜃)
𝑻𝒙𝟏
𝑻𝒙𝟐
𝑻𝒙𝟑

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