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Beamforming.

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Presentation on theme: "Beamforming."β€” Presentation transcript:

1 Beamforming

2 Tx1

3 Tx1 cos⁑(2πœ‹π‘“π‘‘)

4 Tx2 Tx1 cos⁑(2πœ‹π‘“π‘‘) 𝝀 𝟐

5 𝐜𝐨𝐬 πŸπ…π’‡π’• + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝅 Tx2 Tx1 cos⁑(2πœ‹π‘“π‘‘) Rx 𝝀 𝟐

6 Destructive superimposition
Tx2 Tx1 cos 2πœ‹π‘“π‘‘ + cos 2πœ‹π‘“π‘‘+πœ‹ =0 𝝀 𝟐 Zero signal

7 Rx 𝐜𝐨𝐬 πŸπ…π’‡π’• + Tx2 𝝀 𝟐 Tx1

8 Constructive superimposition
cos 2πœ‹π‘“π‘‘ + cos 2πœ‹π‘“π‘‘+0 =2cos⁑(2πœ‹π‘“π‘‘) Amplified signal (twice amplitude) Tx2 𝝀 𝟐 Tx1

9 Receiver at arbitrary location
Rx 𝐜𝐨𝐬 πŸπ…π’‡π’• + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓 Tx2 𝝀 𝟐 Tx1

10 Arbitrary location, what’s the path difference

11 Path difference Rx 𝐜𝐨𝐬 πŸπ…π’‡π’• + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓 πœƒ Tx2 𝒅 Tx1

12 Path difference and phase difference
Rx 𝐜𝐨𝐬 πŸπ…π’‡π’• + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓 πœ™(π‘β„Žπ‘Žπ‘ π‘’ π‘‘π‘–π‘“π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’)= 2πœ‹ πœ† βˆ—(π‘π‘Žπ‘‘β„Ž π‘‘π‘–π‘“π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’) Path difference = 𝒅𝒄𝒐𝒔(𝜽) πœ™= 2πœ‹ πœ† βˆ—π’…π’„π’π’”(𝜽) πœƒ Tx2 𝒅 Tx1 𝐑𝐱 𝜽 =𝐜𝐨𝐬 πŸπ…π’‡π’• + 𝐜𝐨𝐬 πŸπ…π’‡π’•+ πŸπ… 𝝀 𝒅𝒄𝒐𝒔(𝜽)

13 (𝑑= πœ† 2 ) Radiation pattern: Rx amplitude as a function of angle

14 Radiation pattern: Rx amplitude as a function of angle
(𝑑=πœ†)

15 Radiation pattern: Rx amplitude as a function of angle
(𝑑=2πœ†)

16 Radiation pattern: Rx amplitude as a function of angle
(𝑑= πœ† 2 ) Radiation pattern: Rx amplitude as a function of angle 𝐜𝐨𝐬 πŸπ…π’‡π’• + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓

17 Radiation pattern: Rx amplitude as a function of angle
(𝑑= πœ† 2 ) Radiation pattern: Rx amplitude as a function of angle The initial phases can be controlled 𝐜𝐨𝐬 πŸπ…π’‡π’•+ 𝝓 π’Šπ’ + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓

18 Radiation pattern: Rx amplitude as a function of angle
(𝑑= πœ† 2 ) Radiation pattern: Rx amplitude as a function of angle 𝐜𝐨𝐬 πŸπ…π’‡π’•+ 𝝓 π’Šπ’ + 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓 𝝓 π’Šπ’ =0 𝝓 π’Šπ’ =-x A non zero initial phase can change the radiation pattern 𝝓 π’Šπ’ =0 𝝓 π’Šπ’ =-x

19 Multiple antennas

20 . . . Rx πœƒ 𝒅 𝒅 Tx(N) Tx(N-1) Tx2 Tx1 2πœ‹ π‘‘π‘π‘œπ‘ (πœƒ) πœ† 𝑅π‘₯= cos 2πœ‹π‘“π‘‘
…….. + cos 2πœ‹π‘“π‘‘+ π‘βˆ’2 βˆ—πœ™ + cos 2πœ‹π‘“π‘‘+ π‘βˆ’1 βˆ—πœ™

21 𝑅π‘₯= cos 2πœ‹π‘“π‘‘ + cos 2πœ‹π‘“π‘‘+πœ™ +cos⁑(2πœ‹π‘“π‘‘+2πœ™) + ……
𝑅π‘₯= cos 2πœ‹π‘“π‘‘ + cos 2πœ‹π‘“π‘‘+πœ™ +cos⁑(2πœ‹π‘“π‘‘+2πœ™) + …….. + cos 2πœ‹π‘“π‘‘+ π‘βˆ’2 βˆ—πœ™ + cos⁑(2πœ‹π‘“π‘‘+ π‘βˆ’1 βˆ—πœ™) cos 2πœ‹π‘“π‘‘ = 𝑒 𝑖2πœ‹π‘“π‘‘ + 𝑒 βˆ’π‘–2πœ‹π‘“π‘‘ 2 =Re { e i2πœ‹π‘“π‘‘ } 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘+πœ™ + e i2πœ‹π‘“π‘‘+2πœ™ + …….. e i2πœ‹π‘“π‘‘+ π‘βˆ’1 πœ™ + e i2πœ‹π‘“π‘‘+ π‘βˆ’1 πœ™ } 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘ 𝑒 π‘–πœ™ + e i2πœ‹π‘“π‘‘ 𝑒 𝑖2πœ™ + …….. e i2πœ‹π‘“π‘‘ 𝑒 𝑖 π‘βˆ’2 πœ™ + e i2πœ‹π‘“π‘‘ 𝑒 𝑖 π‘βˆ’1 πœ™ } 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ 1 + 𝑒 π‘–πœ™ + 𝑒 𝑖2πœ™ + …….. + 𝑒 𝑖 π‘βˆ’2 πœ™ + 𝑒 𝑖 π‘βˆ’1 πœ™ ) 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ 1βˆ’ 𝑒 π‘–π‘πœ™ 1 βˆ’ 𝑒 π‘–πœ™ } 𝑹𝒙(𝜽)=𝑹𝒆{ 𝐞 π’πŸπ…π’‡π’• πŸβˆ’ 𝒆 π’Šπ‘΅ πŸπ…π’…π’„π’π’”(𝜽) 𝝀 𝟏 βˆ’ 𝒆 π’Š πŸπ…π’…π’„π’π’”(𝜽) 𝝀 }

22 (𝑑= πœ† 2 ) Radiation pattern (𝑁=8) (𝑁=2) (𝑁=4)

23 Rotating the beam 𝑅π‘₯= cos 2πœ‹π‘“π‘‘ + cos 2πœ‹π‘“π‘‘+πœ™ +cos⁑(2πœ‹π‘“π‘‘+2πœ™) + …….. + cos 2πœ‹π‘“π‘‘+ π‘βˆ’2 βˆ—πœ™ + cos⁑(2πœ‹π‘“π‘‘+ π‘βˆ’1 βˆ—πœ™) 𝑅π‘₯= cos 2πœ‹π‘“π‘‘+ πœ™ π‘–π‘›π‘œ + cos 2πœ‹π‘“π‘‘+πœ™+ πœ™ 𝑖𝑛1 +cos⁑(2πœ‹π‘“π‘‘+2πœ™+ πœ™ 𝑖𝑛2 ) + … + cos 2πœ‹π‘“π‘‘+ π‘βˆ’2 βˆ—πœ™+ πœ™ 𝑖𝑛(π‘βˆ’2) + cos⁑(2πœ‹π‘“π‘‘+ π‘βˆ’1 βˆ—πœ™+ πœ™ 𝑖𝑛(π‘βˆ’1) ) 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘+πœ™+ πœ™ 𝑖𝑛0 + e i2πœ‹π‘“π‘‘+2πœ™+ 2πœ™ 𝑖𝑛1 + …….. e i2πœ‹π‘“π‘‘+ π‘βˆ’2 πœ™+ πœ™ 𝑖𝑛(π‘βˆ’2) + e i2πœ‹π‘“π‘‘+ π‘βˆ’1 πœ™+ πœ™ 𝑖𝑛(π‘βˆ’1 } πœ™ π‘–π‘›π‘œ =0, πœ™ 𝑖𝑛1 = πœ™ 𝑖𝑛 , πœ™ 𝑖𝑛2 = 2πœ™ 𝑖𝑛 ……….., πœ™ 𝑖𝑛1 = (π‘βˆ’1)βˆ—πœ™ 𝑖𝑛 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘+πœ™+ πœ™ 𝑖𝑛 + e i2πœ‹π‘“π‘‘+2πœ™+ 2πœ™ 𝑖𝑛 + …….. e i2πœ‹π‘“π‘‘+ π‘βˆ’2 πœ™+ (π‘βˆ’2)πœ™ 𝑖𝑛 + e i2πœ‹π‘“π‘‘+ π‘βˆ’1 πœ™+ (π‘βˆ’2)πœ™ 𝑖𝑛 } Goal is to move the maxima to a different angle theta .. πœ™= 2πœ‹ πœ† βˆ—π’…π’„π’π’”(𝜽) 𝑆𝑒𝑑 πœ™ 𝑖𝑛 =βˆ’πœ™=βˆ’ 2πœ‹ πœ† βˆ—π’…π’„π’π’”(𝜽) 𝑅π‘₯=𝑅𝑒{ e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘ + …….. e i2πœ‹π‘“π‘‘ + e i2πœ‹π‘“π‘‘ } 𝑅π‘₯=𝑅𝑒{ Ne i2πœ‹π‘“π‘‘ } A maxima occurs in the direction of 𝜽

24 Rotating the beam 𝝓 π’Šπ’ =βˆ’π“=βˆ’ πŸπ… 𝝀 βˆ—π’…π’„π’π’”(πŸ”πŸŽ) 𝝓 π’Šπ’ =βˆ’π“=βˆ’ πŸπ… 𝝀 βˆ—π’…π’„π’π’”(πŸ’πŸ“)

25 Networking applications

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)
Tx Path difference = 𝒅𝒄𝒐𝒔(𝜽) πœƒ Rx1 𝒅 Rx2 𝐜𝐨𝐬 πŸπ…π’‡π’•+𝝓 𝐜𝐨𝐬 πŸπ…π’‡π’• πœ™= 2πœ‹ πœ† βˆ—π’…π’„π’π’”(𝜽) 𝜽(𝑨𝒐𝑨)=𝒂𝒄𝒐𝒔 𝝀𝝓 πŸπ…π’…

30 Antenna array . . . Tx πœƒ 𝒅 𝒅 Rx(N) Rx(N-1) Rx2 Rx1 cos 2πœ‹π‘“π‘‘ cos 2πœ‹π‘“π‘‘+πœ™
2πœ‹ π‘‘π‘π‘œπ‘ (πœƒ) πœ†

31 𝑒 𝑖2πœ‹π‘“π‘‘ 𝑒 𝑖0 cos 2πœ‹π‘“π‘‘ 𝑒 𝑖0 𝑅 π‘₯ 1 𝑒 𝑖2πœ‹π‘“π‘‘+πœ™ 𝑒 π‘–πœ™ 𝑅 π‘₯ 2 𝑒 π‘–πœ™ cos 2πœ‹π‘“π‘‘+πœ™ 𝑒 𝑖2πœ‹π‘“π‘‘+2πœ™ 𝑒 𝑖2πœ™ 𝑅 π‘₯ 3 𝑒 𝑖2πœ™ cos 2πœ‹π‘“π‘‘+2πœ™ 𝑒 𝑖2πœ‹π‘“π‘‘ 𝑠 𝑑 = = = = cos 2πœ‹π‘“π‘‘+(π‘βˆ’2)πœ™ 𝑒 𝑖(π‘βˆ’2)πœ™ 𝑅 π‘₯ π‘βˆ’1 𝑒 𝑖2πœ‹π‘“π‘‘+(π‘βˆ’2)πœ™ 𝑒 𝑖(π‘βˆ’2)πœ™ 𝑅 π‘₯ 𝑁 cos 2πœ‹π‘“π‘‘+(π‘βˆ’1)πœ™ 𝑒 π‘–πœ™ 𝑒 𝑖2πœ‹π‘“π‘‘+(π‘βˆ’1)πœ™ 𝑒 π‘–πœ™

32 2πœ‹ π‘‘π‘π‘œπ‘ (πœƒ) πœ† 𝑒 𝑖0 𝑅 π‘₯ 1 𝑅 π‘₯ 2 𝑒 π‘–πœ™ 𝑅 π‘₯ 3 𝑒 𝑖2πœ™ 𝑠 𝑑 = 𝑅 π‘₯ π‘βˆ’1 𝑒 𝑖(π‘βˆ’2)πœ™ 𝑅 π‘₯ 𝑁 𝑒 π‘–πœ™ Steering vector

33 Multiple transmitters
Tx2 Tx1 . . . πœƒ 𝒅 𝒅 Rx(N) Rx(N-1) Rx2 Rx1

34 Multiple transmitters
2πœ‹ π‘‘π‘π‘œπ‘ ( πœƒ 1 ) πœ† 2πœ‹ π‘‘π‘π‘œπ‘ ( πœƒ 2 ) πœ† 2πœ‹ π‘‘π‘π‘œπ‘ ( πœƒ π‘˜ ) πœ† 𝑅 π‘₯ 1 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑅 π‘₯ 2 𝑒 𝑖 πœ™ 1 𝑒 𝑖 πœ™ 2 𝑒 𝑖 πœ™ π‘˜ 𝑒 𝑖2 πœ™ π‘˜ 𝑅 π‘₯ 3 𝑒 𝑖2 πœ™ 1 𝑒 𝑖2 πœ™ 2 𝑠 1 𝑠 2 𝑠 π‘˜ = + + 𝑅 π‘₯ π‘βˆ’1 𝑒 𝑖 π‘βˆ’2 πœ™ 1 𝑒 𝑖 π‘βˆ’2 πœ™ 2 𝑒 𝑖 π‘βˆ’2 πœ™ π‘˜ 𝑅 π‘₯ 𝑁 𝑒 𝑖 (π‘βˆ’1)πœ™ 1 𝑒 𝑖 (π‘βˆ’1)πœ™ 2 𝑒 𝑖 (π‘βˆ’1)πœ™ π‘˜ Output is a linear combination of steering vectors from different directions

35 Multiple transmitters
𝑅 π‘₯ 1 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑠 1 𝑅 π‘₯ 2 𝑒 𝑖 πœ™ 1 𝑒 𝑖 πœ™ 2 𝑒 𝑖 πœ™ π‘˜ 𝑠 2 𝑅 π‘₯ 3 𝑒 𝑖2 πœ™ 1 𝑒 𝑖2 πœ™ 2 𝑒 𝑖2 πœ™ π‘˜ = 𝑅 π‘₯ π‘βˆ’1 𝑒 𝑖 π‘βˆ’2 πœ™ 1 𝑒 𝑖 π‘βˆ’2 πœ™ 2 𝑒 𝑖 π‘βˆ’2 πœ™ π‘˜ 𝑅 π‘₯ 𝑁 𝑒 𝑖 (π‘βˆ’1)πœ™ 1 𝑒 𝑖 (π‘βˆ’1)πœ™ 2 𝑒 𝑖 (π‘βˆ’1)πœ™ π‘˜ 𝑠 π‘˜ Steering Matrix (N x K) K sources (Input Vector) N receivers (Output vector)

36 Detecting AoA of K sources simultaneously

37 𝑅 π‘₯ 1 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑠 1 𝑅 π‘₯ 2 𝑒 𝑖 πœ™ 1 𝑒 𝑖 πœ™ 2 𝑒 𝑖 πœ™ π‘˜ 𝑠 2 𝑅 π‘₯ 3 𝑒 𝑖2 πœ™ 1 𝑒 𝑖2 πœ™ 2 𝑒 𝑖2 πœ™ π‘˜ = 𝑅 π‘₯ π‘βˆ’1 𝑒 𝑖 π‘βˆ’2 πœ™ 1 𝑒 𝑖 π‘βˆ’2 πœ™ 2 𝑒 𝑖 π‘βˆ’2 πœ™ π‘˜ 𝑅 π‘₯ 𝑁 𝑒 𝑖 (π‘βˆ’1)πœ™ 1 𝑒 𝑖 (π‘βˆ’1)πœ™ 2 𝑒 𝑖 (π‘βˆ’1)πœ™ π‘˜ 𝑠 π‘˜

38 Multiply by conjugate of steering vector of source 1
𝑅 π‘₯ 1 𝑒 βˆ’π‘– (π‘βˆ’1)πœ™ 1 𝑒 𝑖0 𝑒 βˆ’π‘– πœ™ 1 𝑒 βˆ’π‘– 2πœ™ 1 .. 𝑒 βˆ’π‘– (π‘βˆ’1)πœ™ 1 𝑒 𝑖0 𝑒 βˆ’π‘– πœ™ 1 𝑒 βˆ’π‘– 2πœ™ 1 .. 𝑒 𝑖0 𝑒 𝑖0 𝑒 𝑖0 𝑠 1 𝑅 π‘₯ 2 𝑒 𝑖 πœ™ 1 𝑒 𝑖 πœ™ 2 𝑒 𝑖 πœ™ π‘˜ 𝑠 2 𝑅 π‘₯ 3 𝑒 𝑖2 πœ™ 1 𝑒 𝑖2 πœ™ 2 𝑒 𝑖2 πœ™ π‘˜ = 𝑅 π‘₯ π‘βˆ’1 𝑒 𝑖 π‘βˆ’2 πœ™ 1 𝑒 𝑖 π‘βˆ’2 πœ™ 2 𝑒 𝑖 π‘βˆ’2 πœ™ π‘˜ 𝑅 π‘₯ 𝑁 𝑒 𝑖 (π‘βˆ’1)πœ™ 1 𝑒 𝑖 (π‘βˆ’1)πœ™ 2 𝑒 𝑖 (π‘βˆ’1)πœ™ π‘˜ 𝑠 π‘˜

39 𝑒 βˆ’π‘– (π‘βˆ’1)πœ™ 1 𝑒 𝑖0 𝑒 βˆ’π‘– πœ™ 1 𝑒 βˆ’π‘– 2πœ™ 1 .. 𝑅 π‘₯ 1 𝑠 1 𝑁 π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ 𝑅 π‘₯ 2 𝑠 2 𝑅 π‘₯ 3 = 𝑅 π‘₯ π‘βˆ’1 𝑅 π‘₯ 𝑁 𝑠 π‘˜

40 𝑅 π‘₯ 1 𝑒 βˆ’π‘– (π‘βˆ’1)πœ™ 1 𝑒 𝑖0 𝑒 βˆ’π‘– πœ™ 1 𝑒 βˆ’π‘– 2πœ™ 1 .. A( πœƒ 1 ) = 𝑅 π‘₯ 2 𝑅 π‘₯ 3 = 𝑠 1 βˆ—π‘+ 𝑠 2 βˆ— π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ + 𝑠 3 βˆ— π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ + …….. 𝑅 π‘₯ π‘βˆ’1 𝑅 π‘₯ 𝑁 All energy from direction πœƒ 1 ( π‘“π‘Ÿπ‘œπ‘š 𝑠 1 ) have been aggregated and amplified

41 𝑅 π‘₯ 1 𝑒 βˆ’π‘– (π‘βˆ’1)πœ™ 2 𝑒 𝑖0 𝑒 βˆ’π‘– πœ™ 2 𝑒 βˆ’π‘– 2πœ™ 2 .. A( πœƒ 2 ) = 𝑅 π‘₯ 2 𝑅 π‘₯ 3 = 𝑠 1 βˆ—(π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’)+ 𝑠 2 βˆ— 𝑁 + 𝑠 3 βˆ— π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ + …….. 𝑅 π‘₯ π‘βˆ’1 𝑅 π‘₯ 𝑁 All energy from direction πœƒ 2 ( π‘“π‘Ÿπ‘œπ‘š 𝑠 2 ) have been aggregated and amplified

42 𝑅 π‘₯ 1 𝑒 βˆ’π‘– (π‘βˆ’1)πœ™ π‘Ÿ 𝑒 𝑖0 𝑒 βˆ’π‘– πœ™ π‘Ÿ 𝑒 βˆ’π‘– 2πœ™ π‘Ÿ .. A( πœƒ π‘Ÿ ) = 𝑅 π‘₯ 2 𝑅 π‘₯ 3 = 𝑠 1 βˆ—(π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’)+ 𝑠 2 βˆ— π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ + 𝑠 3 βˆ— π‘ π‘šπ‘Žπ‘™π‘™ π‘£π‘Žπ‘™π‘’π‘’ + …….. 𝑅 π‘₯ π‘βˆ’1 𝑅 π‘₯ 𝑁 The resultant output is very low .. since multiplied steering vector does not match with any of the incoming signals

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(πœƒ) πœƒ πœƒ 𝑠

44 Detecting multiple AoA
𝑻 𝒙 𝟐 𝑻 𝒙 πŸ‘ A(πœƒ) AoA Spectrum 𝑻 𝒙 𝟏 Suc

45 Close by AoAs cannot be resolved
𝑻 𝒙 πŸ‘ 𝑻 𝒙 𝟐 𝑻 𝒙 𝟏

46 MUSIC algorithm has sharp peaks to resolve close AoA
Based on eigen decomposition and PCA – reference to be provided 𝐴 π‘šπ‘’π‘ π‘–π‘ (πœƒ) 𝑻 𝒙 πŸ‘ 𝑻 𝒙 𝟐 𝑻 𝒙 𝟏

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

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