Adaptive analog beamforming

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ADAPTIVE ANALOG BEAMFORMING
SYED KHALID HUSSAIN
260621774, syed.hussain3@mail.mcgill.ca

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Outline:
 INTRODUCTION
 TYPES OF BEAMFORMING
 ALGORITHMS
 ADAPTIVE ANALOG BEAMFORMING ALGORITHMS
 MODIFIED STRUCTURE AND EXTENDED ALGORITHMS
 RESULTS
 CONCLUSION

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Introduction:
 Beamforming
 Is a spatial filtering technique – ability to enhance energy from a particular
direction while suppressing from the others. [4]
 Analogous to frequency domain analysis of time signals –like frequency selective
filter focus on particular frequency, spatial filtering focus on particular direction.
 Spatial filtering depends on the Weiner filter theory, i.e., to compute statistical
estimate of the unknown random signal using a random signal that contains
some information of it.

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Introduction:
 Applications Of Beamforming Technology:
Applications Description
Radar Phased Array Radar, Synthetic Aperture Radar
Sonar Under Water Source Location And Classification
Communications Directional Transmission and Reception, Smart Antenna System
Imaging Medical Diagnosis and Treatment using Tomography and Ultrasonic
Geophysical
Exploration
Seismic Arrays for Oil Exploration and Detection of Under Ground
Nuclear Test

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Types:
 Types of Beamforming
 Based on the accessibility of Rx signal
to the beamformer, there are two types
I. Digital Beamforming :
 Also known as Multiport Beamforming.
 Beamformer can access signals on all antenna-elements.
 For digitally sampled data, approaches like conjugate-gradient and Least-square
are used for better convergence speed of beamforming algorithms.
 Not very economical as one RF downconverter and ADC has to be used at each
antenna element.
Multi-port Beamforming Architecture

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Types:
Single-port Beamforming Architecture
II. Analog Beamforming :
 Also known as Single-port beamforming.
 Beamformer has no knowledge of signals
on individual antenna-elements.
 For analog data, approaches like gradient
based Perturbation algorithm are used to
update weight.
 Economical as a only one RF
downconverter and ADC is used.

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Algorithms:
 Adaptive Beamforming Algorithms :
Three widely used Digital beamforming algorithms for adaptive estimation of
MSE parameter for the available data.
i. LMS algorithm [5]
• gradient based optimization algorithm employing noisy estimate of
the required gradient vector.
• Stochastic approximation of the steepest decent is used as an
optimization method.
ii. SMI algorithm [5]
• based on the inversion of sample correlation of array processor.
• Optimal weights are the estimates of the Weiner weight solution.
• Used as an alternative to LMS algorithm for rapid convergence.

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Algorithms:
iii. RLS algorithm [5]
• sample-wise updating of the correlation matrix inverse.
• Gradient is estimated by the stochastic approximation of the Gauss-
Newton method.
• Provide better convergence rate, Steady-state MSE over LMS
algorithm.
• Complexity is high, proportional to the square of the number of
unknown weights.

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Adaptive Analog Beamforming Algorithms:
 Assumptions
• Narrowband signal processing, i.e., signals are characterized by single
amplitude and single phase term.
• M antennas receive signals through flat-Rayleigh fading from Q users.
• Users are placed at a far field.
 Signal arrived at the antenna array is given by
𝒙 𝑡 =
𝑞=0
𝑄−1
ℎ 𝑞 𝑖 𝑞 𝑡 𝒔 𝜃 𝑞 + 𝑛(𝑡)
where,
𝑖 𝑞(𝑡) is the transmitted signal from 𝑞 𝑡ℎ
user
ℎ 𝑞 is the channel coefficient
𝒔(𝜃 𝑞) is the steering vector
𝑛(𝑡) is the AWGN noise.

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 Analog Beamforming Algorithms :
Mostly used Analog beamforming algorithms for adaptive estimation of MSE
parameter are:
i. Gradient-based Perturbation Algorithm [1] :
• gradient estimation in single output beamformer is made by perturbing
the weights and correlating the resultant power sequence with the
perturbation sequence.
• Three power measuring perturbation systems are proposed in [1].
 Single receiver perturbation system :
• Perturbation sequence is defined as
𝑺 = 𝜹 1 , 𝜹 2 , … . , 𝜹 𝐾 , K = length of perturbation sequence
• Weight perturbation using perturbation sequence is given as
𝒘+ = 𝒘 + 𝛾𝜹(𝑖), 1 ≤ 𝑖 ≤ 𝐾, 𝛾 is a positive scalar.

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 unbiased MSE gradient estimate, i.e., 2 Rw for a single receiver system is given as
𝑔1 𝒘 =
1
𝛾𝐾
𝑖=1
𝐾
𝑓1(𝒘+, 𝑖)𝜹(𝑖)
where, 𝑓1 𝒘+, 𝑖 is the instantaneous array output power .
 for comparison purpose with perturbation algorithm instantaneous MSE is used, i.e.,
𝑔1 𝒘 ′
=
1
𝛾𝐾
𝑖=1
𝐾
𝑑 𝑖 − 𝒘+
𝐻
𝒙 𝑖 𝜹 𝑖
 weight is updated using equation
𝒘 𝑖 + 1 = 𝑷 𝒘 𝑖 − 𝛼𝒈 𝒘 𝑖 +
𝒔0
𝑀
where, P is projection operator matrix
𝛼 is the step-size
𝒔0 is the steering vector of the desired signal
M is the number of antennas

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• Properties of perturbation sequence
• Orthogonality
• Odd symmetry
 Dual Receiver Perturbation System :
• The weight perturbation and gradient
equation becomes
𝒘+ = 𝒘 + 𝛾𝜹 𝑖 ,
𝒘− = 𝒘 − 𝛾𝜹 𝑖 ,
𝒈2 𝒘 ′
=
1
𝛾𝐾 𝑖=1
𝐾
𝑑 𝑖 − 𝒘+
𝐻
𝒙 𝑖 − 𝒘−
𝐻
𝒙 𝑖 𝜹 𝑖
• Orthogonality
Single Receiver Perturbation System
Dual Receiver Perturbation System

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 Dual Receiver Perturbation with a reference
receiver:
• The weight perturbation and gradient
equation becomes
𝒘+ = 𝒘 + 𝛾𝜹 𝑖 ,
𝒈2 𝒘 ′ =
1
𝛾𝐾 𝑖=1
𝐾
𝑑 𝑖 − 𝒘+
𝐻 𝒙 𝑖 − 𝒘 𝐻 𝒙 𝑖 𝜹 𝑖
• Orthogonality
• Odd symmetry
Dual Receiver Perturbation System
With Reference Receiver

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ii. dmr-based algorithm [3] :
 Estimate the correlations between the Rx signal by using definition of output power
and known desired signal.
𝑃 𝒘 ≝ 𝐸 𝒘 𝐻 𝒙 𝑖 𝒙 𝐻 𝑖 𝒘 = 𝒘 𝐻 𝑹𝒘
𝛼 𝒘 ≝ 𝐸 𝑦 𝑖 𝑑∗
𝑖 = 𝒘 𝐻
𝑧
 Weight is updated using Weiner-Hopf relation
𝑹𝒘 𝑚𝑚𝑠𝑒 = 𝒛
 Estimates of P and α is found by using block of samples of output signal
𝑃 𝒘 𝑚 =
1
𝐾 𝑖= 𝑚−1 𝐾+1
𝑚𝐾
𝑦 𝑖 𝑦∗
𝑖 = 𝒘 𝑚
𝐻
𝒓
𝛼 𝒘 𝑚 =
1
𝐾 𝑖= 𝑚−1 𝐾+1
𝑚𝐾
𝑦(𝑖)𝑑∗
(𝑖) = 𝒘 𝑚
𝐻
𝒛

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where,
𝒘 ≝ 𝒘∗
⊗ 𝒘, ⊗ is the kronecker-product
 weight is updated after 𝑀2
samples
 Regularization of estimated R is done for it’s accurate inversion.
 Dominant-mode-rejection [4] is used as a regularization method and it decomposes
matrix R as follows
𝑹 𝑑𝑚𝑟 =
𝑚=1
𝐽
𝜆 𝑚 𝒒 𝑚 𝒒 𝑚
𝐻 +
𝑚=𝐽+1
𝑀
𝒒 𝑚 𝒒 𝑚
𝐻 𝛼 𝑑𝑚𝑟

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Modified Structure:
 Modified beamforming structure leads to an
extension of single-port algorithms.
 Information at the output of each ADC
is utilized to achieve better performance
and to reduce computations.
 Weight is still updated after 𝑀2samples.
 Autocorrelation values in matrix form are given
by
𝒓 = ( 𝑾 𝐻
)−1
𝑷
Single-port structure with multiple ADCs

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Extended Algorithm:
 Dotted lines in the matrices show 2 ADCs are used, each samples half of the array.
 Estimates of the1st and 4th quadrant of R, i.e., 𝑀2 𝐷 mean-power estimates of main-
diagonal partitions that use information of each individual ADC (where 𝑀2
𝐷2
weights reuse for each partition) can be found by using
𝑃𝑢(𝒘 𝑚[𝑏 𝑢:𝑗 𝑢]) = E 𝑦𝑢 𝑖 𝑦𝑢
∗
𝑖 ,
=
1
𝐾
𝑘= 𝑚−1 𝐾+1
𝑚𝐾
𝑦𝑢 𝑖 𝑦𝑢
∗
𝑖 ,
=
1
𝐾
𝑘= 𝑚−1 𝐾+1
𝑚𝐾
𝒘 𝑚 𝑏 𝑢:𝑗 𝑢
𝐻
𝒙 𝑏 𝑢:𝑗 𝑢
𝑖 ∗ 𝒙 𝑏 𝑢:𝑗 𝑢
𝐻
𝑖 𝒘 𝑚 𝑏 𝑢:𝑗 𝑢
,
where,
𝑢 = 1, … , # of ADCs, 𝑚 = 1, … , 𝑀2
𝐷2
for each 𝑢,
𝑏 𝑢 = (𝑢 − 1) 𝑀 𝐷 + 1 𝑗 𝑢= 𝑢 𝑀 𝐷

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 Mean-power estimates of main-diagonal partitions in matrix form can be written as
Extended Algorithm:
where,
𝑢 = 1, … , # of ADCs, 𝑐 = 𝑀 𝐷 − 1 , 𝑧 𝑢 = (𝑢 − 1) 𝑀2
𝐷 + 𝑐
 For the case of 6 antenna-elements and 2 ADCs,
𝑃1(𝒘1) 𝑃1(𝒘7) 𝑃1(𝒘13)
𝑃1(𝒘2) 𝑃1(𝒘8) 𝑃1(𝒘14)
𝑃1(𝒘3) 𝑃1(𝒘9) 𝑃1(𝒘15)
∆1=
𝑃2(𝒘22) 𝑃2(𝒘28) 𝑃2(𝒘28)
𝑃2(𝒘23) 𝑃2(𝒘29) 𝑃2(𝒘29)
𝑃2(𝒘24) 𝑃2(𝒘30) 𝑃2(𝒘36)
∆2=
 Since the 𝑀2
𝐷 estimates of main-diagonal partitions reuse 𝑀2
𝐷2
weights D
times, therefore the weight structure of 1st weight of each main-diagonal partition,
i.e., 𝒘1 and 𝒘22 can be given as
𝒘1=[𝑤 𝑎, 𝑤 𝑏, 𝑤𝑐, 0,0,0] 𝑇
𝒘22=[0,0,0, 𝑤 𝑎, 𝑤 𝑏, 𝑤𝑐] 𝑇

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Extended Algorithm:
 Remaining 𝑀2
(𝐷 − 1) 𝐷 estimates, i.e., off-diagonal partitions, that use information
of combined ADCs can be found by
𝑃𝑢(𝒘 𝑚) = E (𝑦𝑢 𝑖 + 𝑦𝑣(𝑖))(𝑦𝑢 𝑖 + 𝑦𝑣(𝑖))∗
,
=
1
𝐾
𝑘= 𝑚−1 𝐾+1
𝑚𝐾
(𝑦𝑢 𝑖 + 𝑦𝑣(𝑖))(𝑦𝑢 𝑖 + 𝑦𝑣(𝑖))∗
,
=
1
𝐾
𝑘= 𝑚−1 𝐾+1
𝑚𝐾
𝒘 𝑚 𝑏 𝑢:𝑗 𝑢
𝐻
𝑖 + 𝒘 𝑚 𝑐 𝑢:𝑠 𝑢
𝐻
𝒙 𝑐 𝑢:𝑠 𝑢
𝑖
where,
𝑢, 𝑣 = 1, … , # of ADCs s. t. u ≠ 𝑣 𝑚 = 1, … , 𝑀2
𝐷 ,
𝑏 𝑢 = (𝑢 − 1) 𝑀 𝐷 + 1 𝑗 𝑢= 𝑢 𝑀 𝐷
𝑐 𝑣 = (𝑣 − 1) 𝑀 𝐷 + 1 𝑠 𝑣= 𝑣 𝑀 𝐷

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 Mean-power estimates of off-diagonal partitions in matrix form can be written as
Extended Algorithm:
where,
𝑢, 𝑣 = 1, … , # of ADCs, 𝑧 𝑢 = (𝑢 − 1) (𝑀𝐷 + 𝑀 𝐷 − 1) + (𝑀/𝐷)
 For the case of 6 antenna-elements and 2 ADCs
𝑃1,2(𝒘4) 𝑃1,2(𝒘10) 𝑃1,2(𝒘16)
𝑃1,2(𝒘5) 𝑃1,2(𝒘11) 𝑃1,2(𝒘17)
𝑃1,2(𝒘6) 𝑃1,2(𝒘12) 𝑃1,2(𝒘18)
∆1,2=
𝑃2,1(𝒘19) 𝑃2,1(𝒘25) 𝑃2,1(𝒘31)
𝑃2,1(𝒘20) 𝑃2,1(𝒘26) 𝑃2,1(𝒘32)
𝑃2,1(𝒘21) 𝑃2,1(𝒘27) 𝑃2,1(𝒘33)
∆2,1=

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Extended Algorithm:
 In case, if single antenna-element is left out, say 𝑎 𝑡ℎ
element, then it will be directly
input to 𝑑 𝑡ℎ
ADC and (2𝑀 − 1) mean-power estimates out of 𝑀2
(𝐷 − 1) 𝐷 can be
found by
𝑃𝑑(𝑤 𝑚[𝑎]) = E (𝑦 𝑑 𝑖 𝑦 𝑑
∗
(𝑖)) ,
=
1
𝐾
𝑘= 𝑚−1 𝐾+1
𝑚𝐾
(𝑦 𝑑 𝑖 𝑦 𝑑
∗
(𝑖)) ,
where,
𝑎 = 𝑀 , 𝑑 = 𝐷, 𝑚 = 1, … , 2M − 1
=
1
𝐾
𝑘= 𝑚−1 𝐾+1
𝑚𝐾
𝑤 𝑚 𝑎
∗
𝑥 𝑎 𝑖 ∗ 𝑥 𝑎
∗
𝑖 𝑤 𝑚 𝑎 ,

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 Mean-power estimates of single element can be written as
Extended Algorithm:
 For the case of 7 antenna-elements and 3 ADCs
𝑃3 𝒘7 , 𝑃3 𝒘14 , 𝑃3 𝒘21 , 𝑃3 𝒘28 , 𝑃3 𝒘35 , 𝑃3 𝒘42 , 𝑃3 𝒘43 , …
… 𝑃3 𝒘44 , 𝑃3 𝒘45 , 𝑃3 𝒘46 , 𝑃3 𝒘 𝟒7 , 𝑃3 𝒘48 , 𝑃3 𝒘29
T
∆3=
 Thus, mean-power matrix for the case of 6 antenna-elements and 2 ADCs can be
given as
∆ 𝑢 𝑢 = 1 ∆ 𝑢,𝑣 𝑢 = 2, 𝑣 = 1
∆ 𝑢,𝑣 𝑢 = 1, 𝑣 = 2 ∆ 𝑢 𝑢 = 2

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Extended Algorithm:
 Cross-correlation values are computed separately for each ADC and then estimates
are concatenated for all M elements, i.e., 𝒛 = [ 𝒛1, … , 𝒛 𝐷]
𝑇
𝛼 𝑢 𝒘 𝑚 𝑏 𝑢:𝑗 𝑢
= 𝐸[𝑦 𝑢(𝑖)𝑑∗
(𝑖)],
= 𝐸 𝒘 𝑚 𝑏 𝑢:𝑗 𝑢
𝑑∗ 𝑖 ,
𝒛 𝑢 = (𝑾 𝑚[𝑏 𝑢:𝑗 𝑢] 𝑾 𝑚[𝑏 𝑢: 𝑗 𝑢]
𝐻
)−1 𝑾 𝑚[𝑏 𝑢:𝑗 𝑢] 𝛼 𝑢
where,
𝑢 = 1, … , # of ADCs, 𝑚 = 1, … , 𝑀2
, 𝑏 𝑢 = (𝑢 − 1) 𝑀 𝐷 + 1, 𝑗 𝑢= 𝑢 𝑀 𝐷

24
Extended Algorithm:
 Perturbation algorithm is modified s.t
i. Perturbed weights are applied separately on each set of antennas of ADC
ii. MSE gradient is estimated by using the final output.
iii. Weights are updated using the gradient projection algorithm and applied
on the corresponding antenna element.
𝒘+ 𝑏 𝑢:𝑗 𝑢
𝒘 𝑏 𝑢:𝑗 𝑢
, 𝑖 = 𝒘 𝑏 𝑢:𝑗 𝑢
+ γ𝜹 𝑏 𝑢:𝑗 𝑢
𝑖 ,
𝒈1 𝒘 ′ =
1
𝛾𝐾
𝑖=1
𝐾
𝑑 𝑖 − 𝑦 𝑖 𝜹 𝑖 =
1
𝛾𝐾
𝑖=1
𝐾
|𝑑 𝑖 − 𝒘+
𝐻 𝒙1|𝜹(𝑖) ,
𝒘 𝑖 + 1 = 𝑷 𝒘 𝑖 − 𝛼𝒈 𝒘 𝑖 +
𝒔0
𝑀

25
Results :
 All signals considered as interference are modelled as zero mean, complex Gaussian
rv except the desired signal which is modeled as QPSK signal for 6-element ULA.
 AOA (𝜃0) of the desired signal is set at 𝜋 9, and its variance (σ0
2
) = 1 & σ 𝑁
2
=0.1.
 For interference, four scenarios are considered
 No interference.
 three interfering signals with 𝜃1 = − 𝜋 18, 𝜃2 = 5𝜋 18, 𝜃3 = −2𝜋 9, and σ1
2
=
σ2
2
= σ3
2
= 0.5 respectively.
 Same as (b) except σ1
2
= σ2
2
= σ3
2
= 1.
 one interfering signal with 𝜃1 = − 𝜋 18 & σ1
2
=10.
 Switching behavior of analog components is modelled by introducing a parameter
𝑇 𝑤 , i.e., the finite number of samples required by the weights to switch.
 Quantization behavior of weights is also simulated by keeping the finite number of
bits for analog components.
 step-size for perturbation algorithm is set at 0.0025.

26
Results :
Weight Matrix for dmr
based Algorithm
 Criteria for choosing Weights:
 Weight matrix is full-row rank s.t weights
are linearly independent.
 Weight matrix minimizes the errors of R & z.
 Weight matrix and it’s Kronecker-product is
invertible.

27
Results :
 Comparison of dmr and perturbation based algorithm :

28
Results :
 Comparison of dmr and perturbation based algorithm :

29
Results :
 perturbation based algorithm for three power measuring system:

30
Results :
Weight Matrix for
Extended dmr Algorithm

31
Results :
 Comparison of Extended dmr and perturbation based algorithm :

32
Results :
 Comparison of Extended dmr and perturbation based algorithm :

33
Conclusion :
 Perturbation algorithm is the mostly used single-port beamforming algorithm but it
needs fast switching of weights and high-resolution of analog components.
 Algorithm in [3] overcomes the need of these requirements with even better
performance which makes it a cost effective and promising candidate for high data
rate applications.
 Extension of algorithm in [3] for modified beamforming structure gives advantage of
low computations with better convergence rate and steady state MSE. Performance
enhancement is also noticed for extended perturbation based algorithm when used
with the modified structure.

34
References
[1] A. Cantoni, “Application of orthogonal perturbation sequences to adaptive
beamforming,” Antennas and Propagation, IEEE Transactions on, vol. 28,
no. 2, pp. 191–202, 1980.
[2] L. C. Godara and A. Cantoni, “Analysis of the performance of adaptive
beam forming using perturbation sequences,” Antennas and Propagation,
IEEE Transactions on, vol. 31, no. 2, pp. 268–279, 1983.
[3] B. Lawrence and I. N. Psaromiligkos, “Single-port mmse beamforming,”
in Personal Indoor and Mobile Radio Communications (PIMRC), 2011
IEEE 22nd International Symposium on, pp. 1557–1561, IEEE, 2011.
[4] H. L. Van Trees, Detection, estimation, and modulation theory. John
Wiley & Sons, 2004.
[5] Hema Singh and Rakesh Mohan Jha, “Trends in Adaptive Array
Processing,” International Journal of Antennas and Propagation, vol. 2012, Article
ID 361768, 20 pages, 2012. doi:10.1155/2012/361768

Adaptive analog beamforming

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