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)

The scheme of many inputs, many outputs (Multiple-Input Multiple-Output – MIMO) is becoming more widespread to increase the noise immunity of the transmission of information. In this connection, the need arises for spectral analysis of momentum vectors connected by a MIMO scheme. To obtain the Fourier transform of a four-dimensional (4M) pulse vector, it is proposed to use a quaternion.

Quaternion is a hypercomplex number with one real and three imaginary parts. Quaternion can also be represented in matrix or vector form. The matrix of a single quaternion is orthogonal. The quaternion in the matrix notation is decomposed into basis matrices for which the multiplication rules are the same as for imaginary units. The base matrices of the imaginary part add up to an imaginary matrix. In connection with the noted properties of the quaternion, it is used to obtain the quaternion Fourier transform (QFT). QFT is used to obtain spectra of two-dimensional pulses in the analysis of two-dimensional images. In this paper, we present a method for obtaining QFT 4M pulse vectors used in radio engineering using the matrix representation of a quaternion. Moreover, we do not use imaginary units and operate with real, physically realizable signals. The frequencies of the pulse spectrum are represented as a model in the state space with a state transition matrix in the form of an imaginary matrix of the corresponding frequency. The matrix exponent with the matrix of transition of states to the power of the exponent corresponds to the fundamental matrix, which is a solution of a homogeneous linear differential equation.

It is shown that when using the 4M fundamental matrix, the QFT can be transformed into a one-dimensional integral of the 4M fundamental matrix multiplied by the 4M quaternion vector. When calculating the QFT, we used the right notation of multiplying the vector by the matrix. Examples of calculating the QFT vectors of rectangular and sawtooth pulses are given. The calculation results are presented in the form of a matrix of component spectra when multiplied by a vector of pulse amplitudes, we obtain four pulse spectra of the vectors. 

Using the properties of fundamental matrices and well-known formulas of geometry, an expression is obtained for calculating the vector of spectra shifted by pulses at different times. The expressions obtained make it possible to obtain the spectra of pulse vectors, which can be of various shapes and with different time shifts.

The use of inverse QFT for pulses with a different time shift is shown. The presentation and basic properties of the 4M deltaquaternion or the Dirac delta mass are considered. The quaternionic delta pulse is a pure scalar, which can be represented as a point mass in the 3M coordinate system, and has a filtering property. 

An expression is obtained for the Perceval theorem as applied to the QFT for pulses with different time shifts. For various pulse shifts, the Perceval equality is not always satisfied. The conditions are found under which it is possible to obtain such pulse shifts under which this equality is satisfied.

Sovetov V.M. Quaternion Fourier series of periodic pulse sequence. Radiotekhnika. 2021. V. 85. № 2. P. 83−94.  DOI: 10.18127/j00338486-202102-01 (In Russian).

1. Sovetov V.M. Ispol'zovanie slozhnyh signalov, obrazovannyh ortogonal'nymi operatora-mi, dlya uskoreniya vhozhdeniya v sinhronizm. Radiotekhnika 1989. №4 (In Russian).
2. Sovetov V.M. Optimal'nyj priemnik slozhnyh signalov, obrazovannyh ortogonal'nymi operatorami. Izv. vuzov. Ser. Radioelektronika. 1990. № 4 (In Russian). 
3. Alamouti S.M. A simple transmit diversity technique for wireless communications. IEEE J. Select. Areas Commun. 1998. V. 16.  P. 1451–1458. 
4. I Emre Telatar. Capacity of Multi-antenna Gaussian Channels. Emerging Telecommunications Technologies. 1999. V. 10. Is. 6.  Р. 569–709.
5. Sovetov V.M. Prostranstvennaya fazovaya manipulyaciya kvaternionnogo signala. Special'naya tekhnika. 2016. № 5. S. 7–15  (In Russian).
6. Sovetov V.M., Zhuzhoma V.M., Nazarov O.V. Realizaciya skhemy MIMO s ispol'zovaniem predstavleniya nesushchej dinamicheskoj model'Yu v prostranstve sostoyanij. Uspekhi sovremennoj ra-dioelektroniki 2016. № 8. S. 28–35 (In Russian).
7. Sovetov V.M. Sintez optimal'nogo priemnika giperkompleksnyh signalov. Special'naya tekhnika. 2017. № 2. S. 2–10 (In Russian).
8. Sovetov V.M. Otnositel'naya fazovaya manipulyaciya kvaternionnoj nesushchej. Uspekhi sovremennoj radioelektroniki. 2018. № 1.  S. 20–27 (In Russian).
9. Furman Ya.A. Kompleksnye i giperkompleksnye sistemy v zadachah obrabotki mnogomernyh signalov. M.: Fizmatlit. 2004. 456 s.  (In Russian).
10. Morais J.P. Real Quaternionic Calculus Handbook.
11. Kuhn V. Wireless Communications over MIMO Channels – Applications to CDMA and Multiple An-tenna.
12. Biglieri E. MIMO Wireless Communications. 2007.
13. Le Chung Tran Complex Orthogonal Space-Time Processing in Wireless Communications. 2006.
14. Hitzer E. Quaternion Domain Fourier Transform. International Christian University, Mitaka. Mathe-matics Subject Classification (2000). Primary 11R52; Secondary 42A38, 15A66.
15. Bülow T. Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. Bericht Nr. 9903 August 1999.
16. Guicheng Yang, Yingxiong Fu. Spectrum of Signals on the Quaternion Fourier Transform Domain. Journal of Applied Mathematics and Physics. 2013. № 1. P. 36–38.
17. Ell T.A., Le Bihan N., Sangwine S.J. Quaternion Fourier Transforms for Signal and Image Processing. Wiley-ISTE 2014.
18. Vikas R. Dubey Quaternion Fourier Transform for Colour Images. International Journal of Computer Science and Information Technologies. 2014. V. 5 (3), P. 4411–4416.
19. Krantz S.G. Geometric Function Theory, Explorations in Complex Analysis-Birkhäuser. Boston. 2006.