Utilization of a Unitary Transform for Efficient Computation in the ...

YILMAZER et al.: UTILIZATION OF A UNITARY TRANSFORM FOR EFFICIENT COMPUTATION 179
TABLE I
SUMMARY OF SIGNAL FEATURES INCIDENT ON THE ANTENNA ARRAY
One should state that premultiplying and postmultiplying
by and require only additions and scaling. For
example, in the case of sensor elements, and for a given
pencil parameter , the computations done for the unitary
transformation,
, requires around
real additions. This is negligible as compared to
the computations done for the computation involved in the
eigendecomposition [8], [9].
Eigenstructure based methods for estimating DOA of the
sources impinging on a ULA requires complex calculations
in computing the eigenvectors and the eigenvalues. The MP
method, in addition, requires the computation of a SVD of the
complex-valued data. It should be stated that eigendecomposition
with complex-valued data matrix is quite computation
intensive. The eigen-decomposition process consists of a large
portion of the whole computational load. To reduce the computational
complexity during eigen-decomposition, application
of a unitary transformation is proposed for DOA estimation
by using a real-valued SVD. Computing the eigen-components
of the unitary transformed data matrix requires only real
computations. Real-valued eigendecomposition has a reduced
computational complexity than the complex one approximately
by a factor of four.
The unitary MP (UMP) method is thus a completely realvalued
algorithm as it requires only real-valued computations.
Apart from finding the singular values and vectors, the rest of the
calculations are also real computations as opposed to the ones
done in the conventional MP method. A big portion of the computational
load is occupied by the multiplication operations, so
transforming the data can save a noticeable amount of computations
and the processing time is reduced greatly.
V. SIMULATION RESULTS
In this section, the computer simulation results are given to illustrate
the performance of the UMP method. The noisy signal
model is formulated from (14). is treated as a zero mean
Gaussian white noise with variance . Uniformly spaced arrays
(ULA) of omnidirectional isotropic point sensors are considered
in this study. The distance between any two elements
of the ULA is half a wavelength. is the voltage induced
at each of the antenna elements, for
.Itis
assumed that there are 6 antenna elements and two signals are
impinging on the array with amplitudes
. The
phases, magnitudes and DOA of the two impinging signals are
tabulated in Table I.
Since the estimated direction of arrival is treated as a random
variable, the stability/accuracy of the results need to be expressed
in terms of its statistical properties, which in this case
Fig. 1. Variance 010 log (var( ^ )) UMP, MP, and the CRLB are plotted
against the SNR.
are the estimated values such as the mean, variance, and so
on of the estimate. The Cramer–Rao lower bound (CRLB)
measures the goodness of an estimator. CRLB is the limit that
the variance of the estimates contaminated by white Gaussian
noise cannot be any smaller than this bound. The simulation
results show that the variance of the estimators approaches to
the CRLB. The bound is found by using the Fisher information
matrix ( ), whose diagonal elements are the corresponding
CRLB of that element. Fisher is defined by
(43)
(44)
where is the estimate of . Here, is probability density
function conditioned to an unknown vector parameter . The diagonal
elements of the inverse of the Fisher define the bound
on that estimated value as shown in [15]. In this study, a comparison
in performance is made between the MP and the UMP
method. We compute the CRLB of the variance for the estimate
of the DOA. The inverse of the sample invariance of the estimate
of (DOA) and the inverse variance of the MP method
using complex data, inverse variance of the UMP method and
the corresponding CRLB versus signal-to-noise ratio (SNR) of
the input voltages is plotted in Fig. 1. Different values of SNR
comprise the -axis and the inverse of the variance of the estimated
arrival angle in logarithmic domain,
is shown along the -axis.
To obtain the sample variance of the estimate of , 300
independent trials have been performed. Noise for each
run is independent of each other. In the summary of algorithm
section, the Step 6 implies that all the eigenvalues of
are real. One should
recall that the eigenvalues of a real matrix can either be real
or they occur in complex conjugate pairs. In the case of very
noisy environment, the eigenvalues of this matrix may fail to be
real and they occur in complex conjugate pairs, which is also