Adaptive analog beamforming

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ADAPTIVE ANALOG BEAMFORMING

SYED KHALID HUSSAIN

260621774, [email protected]

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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

where, is the transmitted signal from user is the channel coefficient is the steering vector 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 , K = length of perturbation sequence

• Weight perturbation using perturbation sequence is given as

, , is a positive scalar.

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Adaptive Analog Beamforming Algorithms: unbiased MSE gradient estimate, i.e., 2 Rw for a single receiver system is given as

where, is the instantaneous array output power .

for comparison purpose with perturbation algorithm instantaneous MSE is used, i.e.,

weight is updated using equation

where, P is projection operator matrix

is the step-size

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

• Properties of perturbation sequence• 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

• Properties of perturbation sequence• Orthogonality• Odd symmetry

Dual Receiver Perturbation SystemWith Reference Receiver

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Adaptive Analog Beamforming Algorithms:ii. dmr-based algorithm [3] :

Estimate the correlations between the Rx signal by using definition of output power and known desired signal.

Weight is updated using Weiner-Hopf relation

Estimates of P and α is found by using block of samples of output signal

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where, , is the kronecker-product

weight is updated after 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

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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 samples.

Autocorrelation values in matrix form are given by 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., mean-power estimates of main-diagonal partitions that use information of each individual ADC (where weights reuse for each partition) can be found by using

¿ 1𝐾 ∑

𝑘=(𝑚− 1) 𝐾+1

𝑚𝐾

𝑦𝑢 (𝑖 ) 𝑦𝑢∗ (𝑖 ) ,

where,

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Mean-power estimates of main-diagonal partitions in matrix form can be written as

Extended Algorithm:

where,

For the case of 6 antenna-elements and 2 ADCs,

�̂�1(𝒘 1) �̂�1(𝒘7) �̂�1(𝒘13)

�̂�1(𝒘 2) �̂�1(𝒘8) �̂�1(𝒘1 4)�̂�1(𝒘 3) �̂�1(𝒘9) �̂�1(𝒘15)

∆1=¿�̂�2(𝒘22) �̂�2(𝒘28) �̂�2(𝒘28)

�̂�2(𝒘23) �̂�2(𝒘29) �̂�2(𝒘29)�̂�2(𝒘 24) �̂�2(𝒘30) �̂�2(𝒘36)

∆2=¿

Since the estimates of main-diagonal partitions reuse weights D times, therefore the weight structure of 1st weight of each main-diagonal partition, i.e., and can be given as𝒘1=¿[𝑤𝑎 ,𝑤𝑏,𝑤𝑐 , 0,0,0 ]

𝑇 ¿ 𝒘 22=¿[0,0,0 ,𝑤𝑎 ,𝑤𝑏,𝑤𝑐 ]𝑇 ¿

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Extended Algorithm:

Remaining estimates, i.e., off-diagonal partitions, that use information of combined ADCs can be found by

¿ 1𝐾 ∑

𝑘=(𝑚− 1) 𝐾+1

𝑚𝐾

¿¿

where,

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Mean-power estimates of off-diagonal partitions in matrix form can be written as

Extended Algorithm:

where,

For the case of 6 antenna-elements and 2 ADCs

�̂�1,2(𝒘 4) 𝑃1 ,2(𝒘10) �̂�1, 2(𝒘16)

�̂�1 , 2(𝒘5) 𝑃1 , 2(𝒘11) �̂�1 ,2(𝒘17)�̂�1 ,2(𝒘6) 𝑃1 , 2(𝒘12) �̂�1 ,2(𝒘18)

∆1,2=¿�̂�2,1(𝒘19) �̂�2,1(𝒘 25) �̂�2,1(𝒘3 1)

�̂�2,1(𝒘20) �̂�2,1(𝒘 26) �̂�2,1(𝒘32)�̂�2,1(𝒘21) �̂�2,1(𝒘 27) �̂�2,1(𝒘33)

∆2,1=¿

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Extended Algorithm:

In case, if single antenna-element is left out, say element, then it will be directly input to ADC and mean-power estimates out of can be found by

¿ 1𝐾 ∑

𝑘=(𝑚− 1) 𝐾+1

𝑚𝐾

¿¿

where,

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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

……

T

=

Thus, mean-power matrix for the case of 6 antenna-elements and 2 ADCs can be given as

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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.,

,¿𝐸 [𝒘𝑚 [𝑏𝑢 : 𝑗𝑢]𝒙 [𝑏𝑢: 𝑗𝑢]𝑑

∗ (𝑖 ) ] ,

~̂𝒛𝑢=(𝑾𝑚 [𝑏𝑢 : 𝑗𝑢]𝑾𝑚 [𝑏𝑢: 𝑗𝑢 ]𝐻 )−1𝑾 𝑚[𝑏𝑢 : 𝑗𝑢]𝛼𝑢

where,

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Extended Algorithm:

Perturbation algorithm is modified s.t

i. Perturbed weights are applied separately on each set of antennas of ADCii. 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+ [𝑏𝑢 : 𝑗𝑢 ] (𝒘 [𝑏𝑢 : 𝑗𝑢 ] , 𝑖)=𝒘 [𝑏𝑢 : 𝑗𝑢 ]+γ 𝜹[𝑏𝑢: 𝑗𝑢] (𝑖 ) ,

𝒈1 (𝒘 )′= 1𝛾 𝐾∑

𝑖=1

𝐾

|𝑑 (𝑖 ) − 𝑦 (𝑖 )|𝜹 (𝑖 )= 1𝛾 𝐾 ∑

𝑖=1

𝐾

¿𝑑 (𝑖 )−𝒘+¿𝐻 𝒙 1∨𝜹(𝑖 ) ,¿

𝒘 (𝑖+1 )=𝑷 [𝒘 (𝑖 )−𝛼𝒈 (𝒘 (𝑖 ) ) ]+ 𝒔0

𝑀

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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 of the desired signal is set at , and its variance () = 1 & =0.1.

For interference, four scenarios are considered No interference. three interfering signals with , and respectively. Same as (b) except one interfering signal with & =10.

Switching behavior of analog components is modelled by introducing a parameter 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.

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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.

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Results :

Comparison of dmr and perturbation based algorithm :

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Results :

Comparison of dmr and perturbation based algorithm :

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Results :

perturbation based algorithm for three power measuring system:

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Results :

Weight Matrix for Extended dmr

Algorithm

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Results :

Comparison of Extended dmr and perturbation based algorithm :

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Results :

Comparison of Extended dmr and perturbation based algorithm :

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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.

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References

[1] A. Cantoni, “Application of orthogonal perturbation sequences to adaptivebeamforming,” 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 adaptivebeam 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), 2011IEEE 22nd International Symposium on, pp. 1557–1561, IEEE, 2011.[4] H. L. Van Trees, Detection, estimation, and modulation theory. JohnWiley & 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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