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New Matrix Operations for DSP
Dr. Vadim Slyusar
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Module Status
Lecture
DESCRIPTION
AUTHOR
Dr. Vadim Slyusar
Dr.Vadim Slyusar was born
in Poltava, Ukraine, on
October 15, 1964.
Dr.V.Slyusar has 16 years of
research experience in the
areas of radar systems,
smart antennas for wireless
communications and digital
beamforming. He earned his
Ph.D. in 1992, Dr.D. in 2000
and has 25 patents and 152
publications in these areas.
He is chief of a research
department in the R&D
Group for Electromechanics
and Pulsed Power (Kyiv,
Ukraine), and is an authority
in digital signal processing
for radar applications. You
can contact author at
swadim@profit.net.ua
This lecture presents the basic concepts of new
matrix operations and related applications for digital
beamforming. This lecture can be used for radar
system, smart antennas for wireless communications,
and other systems applying digital beamforming. It's
intended for individuals new to the field who wish to
gain a basic understanding in this area. For additional
information, check out the reference material
presented at the end of this lecture.
PREREQUISITES
Matrix theory and digital beamforming.
INTENDED AUDIENCE
Individuals interested in digital signal processing.
ESTIMATED TIME
30 minutes
View the complete TechOnLine University Course
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The compactness of mathematical models of physical systems;
The best presentation of essence of signal processing algorithms;
Computer time economy.
This matrix is especially advantageous for the digital multichannel systems of data
processing!
Welcome to the TechOnline lecture about the theory of new matrix
operations for digital signal processing.
The application of matrices, as you known, allows us to effeciently
construct a model of a physical system and to formulate the essence of
algorithms for processing signals. Matrix means is especially
advantageous for solving the problems associated with the analysis,
synthesis, capture, and data processing of complex multichannel
systems.
This lecture concentrates on the application of matrices in radar systems
with digital beamforming. However, these matrices also can be utilized
for any system implementing digital beamforming. For example, in
acoustics, hydroacoustics, cellular radio communication, ultrasonic
medical diagnostics, radio astronomy, etc. In addition, these new matrix
procedures can be useful for threedimensional, image visualization
systems.
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The simplest form of radar with digital beamforming is a onecoordinate
Nelements array pattern with an A/D (analogtodigital) converter, with
DSP signal filtering in each channel, and beam sheaf forming with the
help of fast Fourier transforms.
Voltage arrays resulting from beamforming with exposure to signals of M
sources in matrix form, ignoring noise, are written as U = FA, where U is
a vector of complex digital beamformer response voltages. F is an R x M
matrix of the directivity characteristics of R secondary channels at x
coordinates of M sources. A is the vector of complex amplitudes of M
source signals.
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Traditional matrix techniques, used with digital beamforming two
coordinate radars, make it possible to compact the expressions shown
here in (1), which has no unity with the one coordinate model U = PA
we’ve just discussed. The retention of onechannel variant structure
leads to unwieldy expressions or to the introduction of new matrix
operations, such as rows diagonalization or columns diagonalization of
the matrices, as we see here in (2).
When the measuring coordinates are increased, the amount of defects
indicated becomes more evident, and this is a serious problem.
In order to resolve this problem, we offer new operations of matrix
multiplication. This lecture is devoted to their consideration.
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When defining two matrices with an identical amount of lines, the Slyusar
product is a matrix obtained by multipling each element in the left matrix
by a row in the right matrix, which corresponds to a number in the left
matrix.
A symmetrical alternative for the Slyusar product is a transposed Slyusar
product. We’ll consider this next.
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When defining two matrices with an identical number of columns, the
transposed Slyusar product is a matrix obtained by multipling each
element in the left matrix, with a column in the right matrix, which
corresponds to a number in the left matrix.
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Fundamental Properties
Using the Slyusar product allows us to reduce the amount of computing
operations required in widespread problems, when multipling diagonal
matrix A by matrix B, which conforms with it through p X g rows.
For all of this, the transition to the Slyusar product results in a reduction
of p times the multiplication operations, and allows us to completely
exclude the pg(p1) operation in the summation.
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Where xm, ym, wm, and zm is angular coordinates, frequencies and ranges of signals
sources respectively.
As you can see here, the matrix model of a fourcoordinate radar, which
measures the angular coordinates, velocity, and range of M sources, is
obtained with the help of the transposed Slyusar product.
Comparing this model with a matrix model, based on a standard matrix
product, this new model allows us to receive a considerable reduction in the
amount of computing operations. This is very important, for example, for
modeling such systems in the MatLab or MathCad packages.
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This new matrix operation significantly decreases computer time by using
multisignal measurement methods in matrix form. Here, we see a well
know method, maximization likelihood, for reducing the maxmization of
function L.
Calculating the quadratic form by identity is reduced to the Hadamard
product. As a result, for a 32x32 antenna array in each channel with 32
synthesized frequency filters in 32 distance intervals, we can decrease
the amount of multiplication by 8,004 times and the amount of
summations by 8,456 times with respect to the initial notation. In this
case, the number of multiplication operations are decreased by more
than 268.845 billion, as compared to a fourcoordinate model based on a
traditional matrix product.
Next, we’ll consider the CramerRao bound.
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To evaluate the potential accuracy of a maximum likelihood algorithm
measurement, the lower CramerRao bound can be used. This is
obtained by reversing the information in a Fisher matrix. The Fisher
matrix for a fourcoordinate radar is illustrated here, as well as for the
onecoordinate problem. The advantage of these proposed matrix
operations is that the onecoordinate case, by the simple substitution,
can also be used on a multicoordinate case.
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Here’s a Neudecker derivative matrix, in expression form, for the Fischer
information matrix. It’s remarkable that in the case of the Slyusar product,
or its transposed variant, that the result of the Neudecker derivative can
be factorized as shown here. As you can see, we’ve used the symbol of
the modular transposed Slyusar product. Their definitions and properties
are considered next.
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Here, you can see the essence of the modular transposed Slyusar
product.
The modular variants of Slyusar products have specific properties which
we’ll demonstrate next.
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Here, we see a list of the basic properties of modular and modular
transposed Slyusar products.
With the help of the Slyusar product procedure, a mathematical model of
a multisectional array with digital beamforming can be formalized.
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Multisectional Array
The mathematical model of a multisectional array with digital
beamforming is illustrated here. Each section of the array has its own
block of directivity characteristics, frequency, and range characteristics.
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Multistatic Radar System
Here, we see the mathematical model of a multistatic radar system with a
digital beamforming antenna array. In this case, each position of the
radar has its own block of directivity antenna array characteristics,
frequency characteristics, and pulse signal shapes.
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These radar models are predicted based on the availability of the
antenna pattern factorization and identity characteristics of the receiving
channels. When such assumptions are impossible, the response
formalization problem of a flat array can be carried out on the basis of a
new penetrating Slyusar product. It can be determined for any matrix A
and modular matrix B, where the dimensions of the modular matrix are
the same as the dimensions of matrix A.
With the help of a penetrating Slyusar product, it’s possible to record the
response of a threecoordinate radar to a single signal, taking into
account that each receiving channel corresponds to its unique amplitude
frequency characteristic. This mathematical model is illustrated next.
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Digital Beamforming for Nonidentical Channels (1 source)
The response of a threecoordinate flat digital antenna array of R x R
elements can be stated by penetrating the facesplitting product of the
matrices, without noise. Where U denotes a blockmatrix of voltages of
the response channels, A is a complex signal amplitude matrix, or vector
for a single moment of time, Q is a matrix of the directivity characteristics
of the primary channels in azimuth and elevation angle planes, which
cannot be factorized, and F is a blockmatrix of amplitudefrequency
characteristics of T filters for R x R nonidentical reception channels.
For the selection of a single source on four coordinates, azimuth,
elevation angle, frequency, and range, the response of a digital antenna
array can be stated through a generalized Slyusar product or generalized
transposed Slyusar product. We’ll consider them next.
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As you can see, the concepts of the generalized Slyusar product are
illustrated here. An alternative we should also consider is the generalized
transposed facesplitting blockmatrices product. We’ll consider this next.
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Digital Beamforming for Nonidentical Channels (1 source)
The response of a 4coordinate flat digital antenna array of R x R
elements can be stated through a penetrating facesplitting product of
matrices, without noise. Where U denotes a blockmatrix of response
channel voltages, A is a complex signal amplitude matrix, or vector for a
single moment in time, Q is a matrix of the directivity characteristics of
primary channels in azimuth and elevation angle planes that cannot be
factorized, and F is a blockmatrix of amplitudefrequency characteristics
of T filters for R x R nonidentical reception channels, and S is a block
matrix of range characteristics of D range gates.
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Digital Beamforming for Nonidentical Channels (1 source)
Here, we see the mathematical model of a multistatic radar system with a
digital beamforming antenna array for nonidentical channels. In this
case, each position of the radar has its own block of directivity antenna
array characteristics, frequency characteristics, and pulse signal shapes.
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Slyusar V. I. New operations of matrices product for applications of radars, in
Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave
Theory (DIPED97), Lviv, September 1517, 1997, P. 7374 (in Russian).
Slyusar V. I. Analytical model of the digital antenna array on a basis of facesplitting
matrixs products, in Proc. ICATT—97, Kyiv, May 1997.  P. 108 – 109.
Slyusar V. I. The facesplitting matrixs products in radar applications,
Radioelectronics and Communications Systems.  Vol. 41.  no. 3. 1998.
Slyusar V. I. The facesplitting matrixs products family and its characteristics,
Cybernetics and Systems Analysis. Vol. 35. no.3 1999.
Slyusar V. I. On information Fisher matrix for system models bases on face
splitting matrixs products, Cybernetics and Systems Analysis. Vol. 35. no. 4.
1999.
Slyusar V. I. The matrix models of digital antenna arrays with nonidentical
channels, Proc. ICATT 99. Sevastopil. September 811, 1999. pp. 241243.
This concludes our lecture. Here are some references that were used in
this lecture.
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Congratulations, you have now completed
New Matrix Operations for DSP.
Please take a moment to complete a user survey on this course.
You may also print a completion certificate suitable for framing.
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~


Dear Vadim Slyusar:
This document is intended to verify that Vadim Slyusar has submitted
the 1ecture, titled "New matrix operations
for
DSP
" to Tec110 nLine Inc.
TechOnLine has added,
оп
the 9th
of
November 1999, the online version
of
this lecture to its TechOnLine University web site as
ап
educational lecture.
О
п
behalf
of TechOnLine 1want to thank
уо
и
,
Vadim, for all your time
and efforts
о
п
this project.
Sincerely,
.'7
~
Mike Strange
Project
Ma
nager, TechOnLine Inc.
TechOnLine, Inc.
230 Second
А
у
е
п ц
е
,
Suite 105
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МА
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