eigenstructure methods for noise covariance estimation olawoye oyeyele aicip group presentation...
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Eigenstructure Methods for Noise Covariance Estimation
Olawoye Oyeyele
AICIP Group Presentation
April 29th, 2003
Outline
Background
Adaptive Antenna Arrays
Array Signal Processing
Discussion
Next Steps
Objective
Discuss Antenna Arrays and similarities to sensor arrays
Investigate methods used for covariance estimation in adaptive antenna arrays with a focus on applicable eigenstructure methods
Background
Antenna Arrays are a group of antenna elements with signal processing capability which enables the dynamic update of the beam pattern
Various elemental configurations possible:– Linear– Circular– Planar
Major objective is to cancel interference Sensor Arrays are similar to antenna arrays
Uniformly spaced Linear Array
. . . … ..
…
d0 1 3 … k-2 k-1
Signals arriving at the (K-1)th element lag those at the (k-2)th element but lead in time
Adaptive Antenna Array
Generally, complex weights are used.
Basic Antenna Array Parameters
1
0
) sin(N
n
knkdjneAAF
Array Factor: the radiation pattern of the array consisting of isotropic elements
Array Propagation Vector: contains the information on angle of arrival of the signal
wavelengthk
eev dKjkjkdT
and /2
... 1 00 sin)1(sin
Steering Vector
Contains the responses of all the vectors of an array. Used to accomplish electronic Beam Steering – each
element of vector performs phase delay with respect to the next.
In electronic steering no physical movement of the array is done.
Mechanical beam steering involves physically moving the elements of the array.
Multiple steering vectors constitute an Array Manifold– Array manifold is an array of steering vectors
Comparison between Sensor and Antenna Arrays
No. Sensor Arrays Antenna Arrays
1. Multiple sensors’ readouts used to make final decision
Reception of multiple elements are combined to estimate signal
2. Different sensors provide different “views”
Different elements receive multipath* components
3. Not necessarily all sensor readouts are combined
Not necessarily all element receptions are combined
*Multipath components are signal waves arriving at different times because each sample traveled varying distances as a result of reflections.
Array Signal Processing
Techniques employed in adaptive antenna arrays They include:
– Beamforming(Adaptive & Partially Adaptive)– Direction of Arrival Estimation(DOA)
These techniques require the estimation of covariance matrices
Beamforming
Adjusting signal amplitudes and phases to form a desired beam
Estimation of signal arriving from a desired direction in the presence of noise by exploiting the spatial separation of the source of the signals.
Applicable to radiation and reception of energy. May be classified as:
– Data Independent– Statistically optimum– Adaptive– Partially Adaptive
Adaptive Beamforming
Can be performed in both frequency and time domains
Sample Matrix Inversion Least Mean Squares(LMS) Recursive Least Squares(RLS) Neural Network
Two-Element Example
w1 w2
tfjNetI 02)( tfjAetS 02)( 6/
Desired Array Output:
2/0
)()( 212 0 tfj
d Aety
Interference arrives at angle of pi/6
2)2/2(
12 00)( tfjtfj
I NeNety
Received Interference signals:
To completely cancel interference (yd=y) the following weights must be used:
w1=1/2-j/2; w2=1/2+j/2
y
Wiener (Optimal) Solution
Solution Optimum Wiener w
giveswhich
022)})({(
error squared Minimum theComputing
matrix covarianceR
(t)}E{x(t)xR and (t)x(t)}*E{d r where
2)(*)(
)]()(*[)(
1opt
2w
H
22
22
rR
RwrtE
RwwrwtdEtE
txwtdtHH
H
Eigenstructure Technique
For L x L matrix Largest M eigenvalues correspond to M directional
sources L-M smallest eigenvalues represent the background
noise power Eigenvectors are orthogonal – may be thought of as
spanning L-dimensional space
Eigenstructure Technique
The space spanned by eigenvectors may be partitioned into two subspaces– Signal subspace– Noise subspace
The steering vectors corresponding to the directional sources are orthogonal to the noise subspace – noise subspace is orthogonal to signal subspace thus
steering vectors are contained in the signal subspace
When explicit correlation matrix is required it may be estimated from the samples.
Sample Matrix Inversion(SMI)
Operates directly on the snapshot of data to estimate covariance matrix
2
1
2
1
)()(*
)()(
^
^
N
Ni
N
Ni
H
ixid
ixix
r
R
rRw^1^^
Weight Vector can be estimated as:
SMI Disadvantages
Increased computational complexity Inversion of large matrices and numerical instability
due to roundoff errors
Recursive Least Squares(RLS)
N
n
in
N
i
Hin
ixidn
ixixn
r
R
1
~
1
~
)()(*)(
)()()(
where 10 is the forgetting factor – ensures that data in the previous data are forgotten
)()(*)1()(
)()()1()(~~
~~
nxndnn
nxnxnn
rr
RR H
Thus, the matrix is found recursively
Recursive Least Squares(RLS)
Fast convergence even with large eigenvalue spread. Recursively updates estimates
Beam Pattern
Direction of Arrival Estimation(DOA)
DOA involves computing the spatial spectrum and determining the maximas.– Maximas correspond to DOAs
Typical DOA algorithms include:– Multiple SIgnal Classification(MUSIC)– Estimation of Signal Parameters via Rotational Invariance
Techniques(ESPRIT)– Spectral Estimation– Minimum Variance Distortionless Response(MVDR)– Linear Prediction– Maximum Likelihood Method(MLM)
MUSIC is explored in this presentation
MUSIC Algorithm
Useful for estimating– Number of sources– Strength of cross-correlation between source signals– Directions of Arrival– Strength of noise
Assumes number of sources < Number of antenna elements.– else signals may be poorly resolved
Estimates noise subspace from available samples
MUSIC algorithm-contd
)()()( tntAstU
MatrixHermitian denotes H where
]))()())(()([])()([
sidesboth ofn Expectatio theTakingHH
uu tntAstntAsEtutuER
Thus,
Assumes that noise at each array element is additive white and gaussian(AWGN) uncorrelated between elements with the same variance and that arriving signals have a mean of zero.
MUSIC Algorithm-contd
After computing the eigenvalues of Ruu,the eigenvalues of ARssAH can be computing by subtracting the variances as follows:
2nii
If number of incident signals D, is less than number of number of antenna elements M, then M-D eigenvalues are zero.
Spatial Spectrum
2 signals, 8 array elements
Discussion
Signal should lie mostly in subspace spanned by eigenvectors associated with large eigenvalues - noise is weak in this subspace.
Idea of communicating where noise is weak similar to other spectrum optimization problems – e.g. water-filling solution to communication spectrum allocation problem
Signal strength is maximum in subspace where noise is weak
Next steps
Apply to the Restricted Matched Filter problem- select a fixed subset of sensors in a cluster
Obtain results that demonstrate the optimality of the Receiver operating characteristic
References
Lal C. Godara, "Application of Antenna Arrays to Mobile Communications, Part I: Performance Improvements, Feasibility and System Considerations," Proc. of the IEEE, Vol. 85, No 7, pp. 1031- 1060, July 1997.
Lal C. Godara, "Application of Antenna Arrays to Mobile Communications, Part II: Beamforming and Direction of Arrival Considerations," Proc. of the IEEE, Vol. 85, No 8, pp. 1195- 1245, July 1997.
B.D. Van Veen and K. M. Buckley "Beamforming: A versatile Approach to Spatial Filtering" IEEE ASSP Magazine, pp. 4-24, April 1988.
John Litva and Titus Kwok-Yeung Lo, Digital Beamforming in wireless communications, Artech House Publishers, 1996.