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Generalized face-splitting matrix products in models of digital antenna arrays with nonidentical channels

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


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.
... При необходимости совместного оценивания угловых координат и дальностей М источников, в случае неидентичных характеристик приемных каналов ЦАР, решение измерительной задачи нуждается во введении еще одного понятияблочного обобщенного торцевого произведения (БОТП) матриц [81]. Согласно названию, данная операция сводится к выполнению поблочной процедуры обобщенного торцевого произведения. ...
... Напряжения ее выходной импульсной смеси без учета шумов могут быть представлены с помощью операции блочного ОТП в виде [ ] Нетрудно заметить, что особенностью приведенных многосигнальных моделей является представление амплитудного сомножителя в виде кронекеровского произведения вектора амплитуд на единичные матрицы. Для уменьшения количества последних имеет смысл воспользоваться формализацией многосигнальных моделей РЛС на основе операций транспонированных ОТП и БОТП [81]. ...
... Пример 10 [81]. ...
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В монографии изложены в прикладных аспектах современные методологические основы и методический аппарат синтеза средств информационного обеспечения вооружения и военной техники. Описаны варианты использования N-OFDM сигналов для решения задач связи на основе ЦАР, а также теория многокоординатных измерений в РЛС с цифровыми антенными решётками. Материалы монографии могут быть полезны для студентов, аспирантов и докторантов высших технических учебных заведений, а также для научных сотрудников научно-исследовательских и научно-производственных организаций.
... Face-splitting product The Face-splitting product [30] (or transposed Khatri-Rao product) is defined as the rows-by-rows tensor product: ...
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We construct a matrix $M\in R^{m\otimes d^c}$ with just $m=O(c\,\lambda\,\varepsilon^{-2}\text{poly}\log1/\varepsilon\delta)$ rows, which preserves the norm $\|Mx\|_2=(1\pm\varepsilon)\|x\|_2$ of all $x$ in any given $\lambda$ dimensional subspace of $ R^d$ with probability at least $1-\delta$. This matrix can be applied to tensors $x^{(1)}\otimes\dots\otimes x^{(c)}\in R^{d^c}$ in $O(c\, m \min\{d,m\})$ time -- hence the name "Tensor Sketch". (Here $x\otimes y = \text{asvec}(xy^T) = [x_1y_1, x_1y_2,\dots,x_1y_m,x_2y_1,\dots,x_ny_m]\in R^{nm}$.) This improves upon earlier Tensor Sketch constructions by Pagh and Pham~[TOCT 2013, SIGKDD 2013] and Avron et al.~[NIPS 2014] which require $m=\Omega(3^c\lambda^2\delta^{-1})$ rows for the same guarantees. The factors of $\lambda$, $\varepsilon^{-2}$ and $\log1/\delta$ can all be shown to be necessary making our sketch optimal up to log factors. With another construction we get $\lambda$ times more rows $m=\tilde O(c\,\lambda^2\,\varepsilon^{-2}(\log1/\delta)^3)$, but the matrix can be applied to any vector $x^{(1)}\otimes\dots\otimes x^{(c)}\in R^{d^c}$ in just $\tilde O(c\, (d+m))$ time. This matches the application time of Tensor Sketch while still improving the exponential dependencies in $c$ and $\log1/\delta$. Technically, we show two main lemmas: (1) For many Johnson Lindenstrauss (JL) constructions, if $Q,Q'\in R^{m\times d}$ are independent JL matrices, the element-wise product $Qx \circ Q'y$ equals $M(x\otimes y)$ for some $M\in R^{m\times d^2}$ which is itself a JL matrix. (2) If $M^{(i)}\in R^{m\times md}$ are independent JL matrices, then $M^{(1)}(x \otimes (M^{(2)}y \otimes \dots)) = M(x\otimes y\otimes \dots)$ for some $M\in R^{m\times d^c}$ which is itself a JL matrix. Combining these two results give an efficient sketch for tensors of any size.
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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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The paper considers a data transmission method in radio relay communication systems based on the use of nonorthogonal frequency discrete modulation (N-OFDM) of signals in combination with their orthogonal polarization. Synthesis of demodulation procedure of N-OFDM signals is conducted with regard for the presence of cross-polarization interference. In order to analyze the potentialities of the procedure synthesized, it is proposed to employ the well-known procedure of calculating the Cramer-Rao lower bound for variance of potentially achievable errors at measurement of quadrature components of signal amplitudes.
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A new principle is suggested for shaping pulse signals in the transmitting antenna of the MIMO-system. The new method differs from the known ones in introducing, in every channel, a certain time shift of signals. As a result, in space there occurs superposition of pulses overlapping in time. At the reception side, after analog-digital conversion of the signal mixture, by known times of signal arrival we can estimate their amplitude components, and perform demodulation of the transmitted messages.
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