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New Matrix Operations for DSP

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  • Central Scientific Research Insitute of Armaments and Military Equipment of Armed Forces of Ukraine

Abstract

This lecture presents the basic concepts of a lot of matrix operations and related applications for digital beamforming, which was proposed by author in 1996-1998. 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.
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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
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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 three-dimensional, image visualization
systems.
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The simplest form of radar with digital beamforming is a one-coordinate
N-elements array pattern with an A/D (analog-to-digital) 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
weve just discussed. The retention of one-channel 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. Well 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
The fundamental properties of these proposed matrix operations are
presented here. Well discuss each of them in more detail next.
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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(p-1) 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 four-coordinate 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 four-coordinate model based on a
traditional matrix product.
Next, well consider the Cramer-Rao bound.
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To evaluate the potential accuracy of a maximum likelihood algorithm
measurement, the lower Cramer-Rao bound can be used. This is
obtained by reversing the information in a Fisher matrix. The Fisher
matrix for a four-coordinate radar is illustrated here, as well as for the
one-coordinate problem. The advantage of these proposed matrix
operations is that the one-coordinate case, by the simple substitution,
can also be used on a multi-coordinate case.
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Heres a Neudecker derivative matrix, in expression form, for the Fischer
information matrix. Its 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, weve used the symbol of
the modular transposed Slyusar product. Their definitions and properties
are considered next.
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Here, we see an illustration of these modular Slyusar product concepts.
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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
well 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, its possible to record the
response of a three-coordinate 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 three-coordinate flat digital antenna array of R x R
elements can be stated by penetrating the face-splitting product of the
matrices, without noise. Where U denotes a block-matrix 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 block-matrix of amplitude-frequency
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. Well 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 face-splitting block-matrices product. Well consider this next.
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This is the definition of a generalized transposed Slyusar product and an
example of its application.
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Digital Beamforming for Nonidentical Channels (1 source)
The response of a 4-coordinate flat digital antenna array of R x R
elements can be stated through a penetrating face-splitting product of
matrices, without noise. Where U denotes a block-matrix 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 block-matrix of amplitude-frequency 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 non-identical 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 (DIPED-97), Lviv, September 15-17, 1997, P. 73-74 (in Russian).
Slyusar V. I. Analytical model of the digital antenna array on a basis of face-splitting
matrixs products, in Proc. ICATT97, Kyiv, May 1997. - P. 108 109.
Slyusar V. I. The face-splitting matrixs products in radar applications,
Radioelectronics and Communications Systems. - Vol. 41. - no. 3. -1998.
Slyusar V. I. The face-splitting 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 8-11, 1999. -pp. 241-243.
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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Vadim Slyusar
New Matrix Operations for DSP
On this date of 04/14/2002
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... The Khatri-Rao product is useful in variances in statistics, multi-way models, linear matrix equations, and signal processing [26][27][28][29][30]. The face-splitting product is useful in convolutional layers in neural networks and digital signal processing in a digital antenna array [31,32]. ...
Full-text available
Preprint
Tensor algebra lies at the core of computational science and machine learning. Due to its high usage, entire libraries exist dedicated to improving its performance. Conventional tensor algebra performance boosts focus on algorithmic optimizations, which in turn lead to incremental improvements. In this paper, we describe a method to accelerate tensor algebra a different way: by outsourcing operations to an optical microchip. We outline a numerical programming language developed to perform tensor algebra computations that is designed to leverage our optical hardware's full potential. We introduce the language's current grammar and go over the compiler design. We then show a new way to store sparse rank-n tensors in RAM that outperforms conventional array storage (used by C++, Java, etc.). This method is more memory-efficient than Compressed Sparse Fiber (CSF) format and is specifically tuned for our optical hardware. Finally, we show how the scalar-tensor product, rank-$n$ Kronecker product, tensor dot product, Khatri-Rao product, face-splitting product, and vector cross product can be compiled into operations native to our optical microchip through various tensor decompositions.
Full-text available
Conference Paper
Under consideration of multi coordinate digital antenna arrays (DAA) there is the problem of compact matrix record of the reception channels. The known mathematical apparatus does not allow to use habitually for the perception structures of matrixes, describing directivity characteristics of the antenna elements and amplitude-frequency characteristics of the filters. For the solution of the given problem, it is offered to operate with a special type of the product of matrixes, named by the author as "face-splitting".
Full-text available
Article
At the present time, new matrix products are suggested to be used in signal processing in the digital antenna arrays (DAA). In this work, the new matrix operations are suggested for compact response recording in radio engineering systems using the digital formation technology of antenna arrays directional patterns in nonidentical receiving channels. The face-splitting matrix products are applied to solving the problems of radar location and communication on the basis of the DAA.
Full-text available
Article
This article continued the description of properties of the face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996.
Full-text available
Article
Expressions for blocks of the information Fisher matrix are presented based on factorization of the Neudecker derivative of a transposed face-splitting matrix product.
Full-text available
Conference Paper
The new concept of face-splitting and transposed face-splitting matrix products is determined; its main characteristics and modifications of the new types of products for module matrices are considered
The face-splitting matrixs products in radar applications, Radioelectronics and Communications Systems
  • V I Slyusar
Slyusar V. I. The face-splitting matrixs products in radar applications, Radioelectronics and Communications Systems. -Vol. 41. -no. 3. -1998.
The face-splitting matrixs products family and its characteristics// Cybernetics and Systems Analysis
  • V I Slyusar
l Slyusar V. I. The face-splitting matrixs products family and its characteristics// Cybernetics and Systems Analysis.-Vol. 35.-no.3 -1999.
On information Fisher matrix for system models bases on facesplitting matrixs products.// Cybernetics and Systems Analysis
  • V I Slyusar
l Slyusar V. I. On information Fisher matrix for system models bases on facesplitting matrixs products.// Cybernetics and Systems Analysis.-Vol. 35.-no. 4. -1999.
The matrix models of digital antenna arrays with nonidentical channels// Proc. ICATT -99.-Sevastopil
  • V I Slyusar
l Slyusar V. I. The matrix models of digital antenna arrays with nonidentical channels// Proc. ICATT -99.-Sevastopil.-September 8-11, 1999. -pp. 241-243.