V.M. Sovetov

16 Central Research Testing Order of the Red Star Institute Ministry of Defense of the Russian Federation 

 n. a. Marshal of the Communications Forces A.I. Belov (Moscow, Russia)

Formulation of the problem. Currently, the multi-input multi-output (MIMO) scheme is widely used to improve the noise immunity of reception. In this case, the information sequences of pulses form a vector, the elements of which are converted by the channel matrix into the transmitted vector. To analyze MIMO channels, it is necessary to have an appropriate technique for obtaining the spectra of the pulse vector.

Target. Propose a technique for calculating the quaternionic Fourier series of a periodic sequence of pulses.

Practical significance. A technique for calculating the Fourier series is presented, in which a sequence of four pulses in time is written in the form of an analytical hypercomplex signal with one real part and three imaginary ones. The quaternion is represented in matrix extension, and the sequence of pulses in vector. The quaternion signal is the output of the model in the state space, while the amplitude and sign of the pulses are determined by the vector of initial states. The calculation results are presented in the form of a matrix related spectral coefficients of the elements of the sequence of pulses, when multiplied by the vector of initial states, we obtain the spectral coefficients of the expansion of pulses in the Fourier quaternion series.

Sovetov V.M. Quaternion Fourier series of periodic pulse sequence. Radiotekhnika. 2021. V. 85. № 3. P. 5−15.  DOI: https://doi.org/10.18127/j00338486-202103-01 (In Russian).

1. Советов В.М., Жужома В.М., Назаров О.В. Реализация схемы MIMO с использованием представления несущей динамической моделью в пространстве состояний // Успехи современной радиоэлектроники. 2016. № 8. С. 28−35.
2. Советов В.М., Жужома В.М., Назаров О.В. Синтез пространственно-временного кода в пространстве состояний с использованием кронекерова произведения //Успехи современной радиоэлектроники. 2016. № 12. С. 29−37.
3. Советов В.М. Пространственная фазовая манипуляция кватернионного сигнала //Специальная техника. 2016. № 5. С. 7−15.
4. Советов В.М. Синтез оптимального приёмника гиперкомплексных сигналов // Специальная техника. 2017. № 2. С. 2−10.
5. Советов В.М. Относительная фазовая манипуляция кватернионной несущей // Успехи современной радиоэлектроники. 2018. № 1. С. 20−27.
6. Ell Todd A., Le Bihan Nicolas, Sangwine Stephen J. Quaternion Fourier Transforms for Signal and Image Processing. Wiley-ISTE 2014.
7. Eckhard Hitzer Quaternion Domain Fourier Transform. International Christian University. Mitaka. Mathematics Sub ject Classification (2000). Primary 11R52. Secondary 42A38, 15A66.
8. Guicheng Yang, Yingxiong Fu. Spectrum of Signals on the Quaternion Fourier Transform Domain // Journal of Applied Mathematics and Physics. 2013. № 1. Р. 36−38.
9. Vikas R. Dubey Quaternion Fourier Transform for Colour Images // International Journal of Computer Science and Information Technologies. 2014. V. 5. № 3. Р. 4411−4416.
10. Фурман Я.А. Комплексные и гиперкомплексные системы в задачах обработки многомерных сигналов. М.: ФИЗМАТЛИТ. 2004. 456 с.