Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa

Universidade de BrasíliaLaboratório de Processamento de Sinais em Arranjos 1

Adaptive & Array Signal ProcessingAdaptive & Array Signal ProcessingAASPAASP

Prof. Dr.-Ing. João Paulo C. Lustosa da CostaUniversity of Brasília (UnB)

Department of Electrical Engineering (ENE)Laboratory of Array Signal Processing

PO Box 4386Zip Code 70.919-970, Brasília - DF

Homepage: http://www.pgea.unb.br/~lasp

Universidade de BrasíliaLaboratório de Processamento de Sinais em Arranjos

Eigenfilter (1)Eigenfilter (1)

We assume that the signal samples u(n) and the noise samples v(n) are uncorrelated.

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In the noiseless case, the average power output of a linear filter is given by

Without signal component, we have that the average power output is

Therefore, the SNR is given by


Eigenfilter (2)Eigenfilter (2)

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We desire to maximize the SNR with the following constraint

Therefore:

The expression is maximized when w is the eigenvector corresponding to the greatest eigenvalue.

The expression is maximized when w is the eigenvector corresponding to the greatest eigenvalue.


Wiener Filters (1)Wiener Filters (1)

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Our filter output is given by

We desire to find a certain y(n), which is an estimate of the desired signal d(n). Therefore, we can define the estimation error e(n).

We define the cost-function as the mean-square error

Computing the gradient of the cost-function



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Computing the gradient of the cost-function



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In the optimum case, the inputs should be orthogonal to the error function.

The principle of the orthogonality. This property allows to check if the linear filter is

operating in its optimum condition. As a consequence:

In the optimum case:



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(I) In the optimum case, the inputs should be orthogonal to the error function.

(II) ) In the optimum case, the ouputs should be orthogonal to the error function. Geometric interpreation



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Wiener-Hopf Equations

Using the definition of estimation error



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Wiener-Hopf Equations

Using the correlation matrix R and the cross-correlation vector p

The optimum tap weight vector is given by



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Linearly Constraint Minimum Variance (LCMV) Filter

Consider some sinusoidal signal

We assume a certain linear constraint with a complex gain g

Primal equation



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The power output is given by



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We desire to maximize the power output considering the constraint. Therefore, we can apply the Lagrange multipliers.



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In the matrix form:

Therefore:

Replacing in the primal equation:



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The Langrage multiplier is given by

Also known as LCMV beamformer



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Comparison LCMV versus Delay and Sum

In delay and sum, the weight vector depends on only one parameter



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Spatial Power Spectrum with DS – without noise



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Spatial Power Spectrum with LCMV without noise



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Spatial Power Spectrum with DS – with noise



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Computing Weight Power |w|2 for LCMV with g = 1



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Spatial Power Spectrum with LCMV with noise



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Spatial Power Spectrum - LCMV with noise and with restriction |w|2 = 1



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CAPON: Minimum Variance Distortionless Response Beamformer (MVDR)



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Spatial Power Spectrum – CAPON with noise



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CAPON belongs to the class of high resolution or super resolution signal processing schemes It surpasses the limiting behavior of the Fourier-based methods (for

instance DS).



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Spatial Power Spectrum – DS with noise, two sources and M = 10



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Spatial Power Spectrum – CAPON with noise, two sources and M = 10



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Spatial Power Spectrum – DS with noise, two sources and M = 7



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Spatial Power Spectrum – CAPON with noise, two sources and M = 7



Direction of arrival (DOA) estimation Planar wave front: depends on the distance and on the array size Narrowband signal → Linear mixture

In this example: Model order d = 4 Number of sensors M = 5

Assuming that N > M, and since M > d. Here we have an overdetermined problem.



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Direction of arrival (DOA) estimation

M-1

3

2

1

0

d

Relation between DOA and the spatial frequency

If the signal has = 0, then the phase shift between the outputs is also zero.

Only one source, i.e. d = 1.



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Multiple Signal Classification (MUSIC) Assuming that the model order is known and equal to d, we compute

the EVD of the correlation matrix.

Knowing the model order, we can separate the correlation matrix into signal and noise parts.

The cost-function to be maximized is the following



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Multiple Signal Classification (MUSIC) MUSIC achieves a precision greater than CAPON, since it takes into

account the data structure.

Documents

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa