Direction of arrival (DOA) is important in many sensor systems such as radar, sonar, electronic surveillance, seismic exploration, and personal communication.

Most of high resolution direction of arrival methods are depend on the eigenvalues and eigenvectors of the received signal.

So the principal components analysis (PCA) neural network is used to extract the eigenvalues and eigenvectors.

The PCA neural network has an advantage over the QR, and power methods in that, its extract the eigenvalues and eigenvectors directly from the received signal without the need to compute the covariance matrix like the QR and power methods.

The Objective of this work is to use the signal subspace techniques to estimate DOA of the received signal with the help of the PCA neural network.

The signal subspace techniques are algorithms for estimation the direction of arrival. Which is a type of a parametric method for DOA estimation. Where the parametric method are classified as shown innext figure

Parametric Method

Spectral Based

SML

MUSIC

CaponConventional

ESPRIT

Subspace Based

Beamformer Technique

DML

Parametric Based

The conventional beamformer is a natural extension of classical Fourier based spectral analysis to sensor data. Its power spectrum is

PBF =aH(θ)Ŕa(θ)/ (aH(θ)a(θ))

Where a(θ) is the steering matrix, and Ŕ is the covariance matrix

Its attempt to resolving the power of two sources spaced closer than beamwidth. Its power spectrum is

PCAP=1/(aH(θ)R-1a(θ))

It’s the first method that dedicated to DOA estimation, which

is depend on the noise subspace. Its power spectrum is

PM=aH(θ)a(θ)/(aH(θ)∏ a(θ)

Where ∏ =UnUnH, Un is the noisy eigenvectors.

Data Matrix

X

L

N

= Eigendecomposition

Computes

)LS, or, TLS(

^^

U

H

U^

H

s

^

UH

nU s

^

U

1

^

U

2

^

U

)/)(arg(cosm

1m

^

T T 1

1

^

2

^

UU

Its required an extensive amounts of formulation and matrix manipulation, thus the block diagram below illustrate it.

If we take the signal covariance matrix RS instead of all the covariance matrix R (i.e., ignoring the noisy covariance matrix Rn ) of the Capon method then its power spectrum will be

PMCAP=1/(aH(θ)RS-1a(θ))

DOA (deg) for the MCapon method.DOA (deg) for the MUSIC method.DOA (deg) for the Capon method.

And for the ESPRIT algorithm is θ1=79.475º , θ2= 82.636º.

Figures below shows the resolution of these algorithms for θ1=80º, θ2=82º, and a ULA of ten sensors

From above figures it can be seen that the MCapon method give higher resolutions than the other methods.

The above algorithms need the number of the received signals. Thus many

methods are used for estimation the number of signals such as the AIC, MDL,

and their modification OSAIC, and OSMDL.

The PCA NN is used with unsupervised algorithms such as:

1. Symmetric Subspace Algorithm

2. Generalized Hebbian algorithm (GHA)

3. Adaptive Principal Components Extraction Algorithm (APEX)

4. Cascade Recursive Least Square Algorithm (CRLS)

ULA or UCA

PCA

Estimation of the number of the

received signals

DOA Estimation

Output angle

The proposed model for DOA estimation is shown in figure below

Where the first block represents the geometry of sensors, which is either a uniform linear array or a uniform circular array as shown in figure below

x0 d 2d )L-1(d

. . .

y

Uniform linear array geometry.

xR

2/Lθ

Ф

z

y

Uniform circular array geometry.

Ith plane wave

The second block represents the PCA neural network with complex algorithms to extract the principal components of the impeding signals on the ULA or UCA, that in turn are used by the third block for estimating the number of sources.

The fourth block represents the subspace algorithms for DOA estimation.

A single source of energy that illuminating a UCA of eight

sensors will be presents. The source with elevation angle θ=30º,

azimuth angle Ф=50º, fm/fS=.1, and SNR=10 db.

When training the PCA neural network with the GHA, APEX, CRLS

algorithms and with an adaptive learning rate then after the

synaptic weights reach their steady state as shown in figures below,

and after satisfying the orthogonality condition that is

W1W1H=1 for GHA algorithm

W1W1H=1 for APEX algorithm

W1W1H=1 for CRLS algorithm

Change of the synaptic weights versus the number of iterations for the complex GHA with adaptive learning rate, j=1, 2, . . ., 8.

Change of the synaptic weights versus the number of iterations for the complex APEX with learning rate=.01, j=1, 2, . . . ,8.

Change of the synaptic weights versus the number of iterations for the complex CRLS with learning rate=.01, j=1, 2, . . . ,8.

Then for j=1, 2, …,8, where yj, Wj is the output and the synaptic weights of the PCA neural network, and λj, and qj is the eigenvalues and eigenvectors.

jjjqand,y

j W

%03.981008

1i

i

1

%76.851008

1i

i

1

%53.921008

1i

i

1

For GHA algorithm

For CRLS algorithm

For APEX algorithm

Hence the first eigenvalue and its eigenvectors represents the principal components. In the other hand there is only one source.

42% %100R/wwyyRs

H

11

*

11s For GHA algorithm

50.11% %100R/wwyyRs

H

11

*

11s

10.45% %100R/wwyyRs

H

11

*

11s For CRLS algorithm

For APEX algorithm

From the eigenvalues, and eigenvectors of GHA, APEX, and CRLS it can be seen that

Also from the eigenvalues and eigenvectors of these algorithms it can be seen that

from above it can be seen that the GHA, and APEX give high reconstruction error, while the CRLS give acceptable reconstruction error. This is because the synaptic weights tend to deconverge the calculation of the principal components of small eigenvalues.

DOA using the principal component of the complex CRLS algorithm.

DOA using the principal component of the complex APEX algorithm.

DOA using the principal component of the complex GHA algorithm.

Now RS can be computed RS=y1y1W1WH

1

Then applied RS to MCapon method as shown in figures below

It is easy to see the effect of the noise on the DOA of both the GHA, and APEX

algorithms due to the high reconstruction error, while the CRLS gives a correct

DOA.

The power full of the CRLS algorithm is that the extraction of the

principal components is carried out from the error vector, i.e., not

directly from the input vector as in the GHA and APEX algorithm.

From the studied cases, and the simulation results presented in this work, the following conclusions can be pointed as follows:

1. A modification to the Capon method has been presented, which give higher resolution.

2. The numbers of sources are computed directly from the output of the PCA neural network instead of using the Maximum Description Length (MDL), Akaike's information criteria (AIC), order statistic maximum description length (OSMDL), or order statistic Akaike's information criteria (OSAIC ) algorithms.

3. An on-line unsupervised learning algorithm for extracting of complex valued principal components such as complex generalized Hebbian algorithm, complex adaptive principal extraction, complex cascade recursive least square have been derived.

5. This work can be especially useful for nonstationary signals, i.e. in the case of the updating of the eigenvectors is slow for the new arriving samples.

4. The principal components are computed directly from the input signals instead of computing it from the covariance matrix.

6. The maximum number of signals that can be estimated is less or equal to the number of sensors. That is, the DOA can not be estimated if the number of signals is greater than the number of sensors.

Hardware implementation for the PCA neural network using filed programmable gate array (FPGA), which gives a fast multiplication, leading to fast updating of the neural network.

Using a technique to estimate the DOA, frequency, and velocity. Such as the maximum likelihood method.

Using the nonparametric method that depends on the fast Fourier transform, and the wavelet transform for the estimation of the DOA.