# A Method for the Design of Bicomplex Orthogonal DSP Algorithms for Applications in Intelligent Radio Access Networks

^{*}

## Abstract

**:**

## 1. Introduction

^{n}, where n denotes the number of imaginary units in the particular numerical system. For n = 0, the one-dimensional (1D) real numbers ( —Real) are obtained. For n = 1 we get the two-dimensional (2D) complex numbers ( —Complex). Thereafter, n = 2 gives rise to the four-dimensional (4D) quaternions ( —Quaternions). According to the logic of doubling the dimensions on which Cayley–Dickson algebra is built, after quaternions the so-called 8D octonions (

**𝕆**—Octonions) can be obtained, followed by 16D sedenions, 32D pathions, 64D chingons, 128D routons, and 256D voudons. All numerical systems with a dimension higher than the second are called hypercomplex numbers. The value of the square of the imaginary units and the interdependencies they obey determine the different types of arithmetic contained in each numerical system. For example, 1D numbers can be integers or fractions, while 2D numbers can be classic complex numbers, hyperbolic numbers, or dual-complex numbers.

**𝕆**) have found practical applications so far. In addition to numerous practical solutions based on real or complex numbers, intensive research based on 4D and 8D numbers has been conducted over the past three or four decades. The applications of the mathematically complicated but very powerful 4D hypercomplex numbers are both copious and ubiquitous. Among them are areas such as robotics, space technology, telemedicine, air and sea navigation, seismology, meteorology, microbiology, geology, and many others. Octonions are not as well studied as complex numbers and quaternions; hence, they have not yet been widely used in practice. Some current areas of application of 8D octonions include string theory, special relativity theory, quantum logic, and the entropy and thermodynamics of black holes.

## 2. 4D Numerical Systems Applicable to DSP

_{S}= a and Q

_{V}= bi

_{1}+ ci

_{2}+ di

_{3}are called the scalar and vector parts of the quaternion, respectively, while i

_{1}, i

_{2}, and i

_{3}are mutually orthogonal imaginary units whose squares, in general, can be equal to −1, +1, or 0. Depending on these numerical values, as well as the multiplication rules that i

_{1}, i

_{2}, and i

_{3}comply with, different types of quaternions are defined, such as pseudo-quaternions, degenerate quaternions, and degenerate pseudo-quaternions [30]. These and many others belong to the large family of 4D hypercomplex numbers. The most commonly used are Hamilton’s quaternions, which are often simply called quaternions. The three imaginary units, known as the orthogonal basis of a quaternion, obey the following rules:

_{1}and C

_{2}, with specifically defined properties, as follows [31]:

_{1}

^{2}= {−1, 0, +1}, i

_{2}

^{2}= {−1, 0, +1} as well as i

_{1},i

_{2}and i

_{1}i

_{2}are uncorrelated and commutative (i

_{1}i

_{2}= i

_{2}i

_{1}) imaginary units. Therefore, bicomplex numbers are complex numbers with complex coefficients—hence, the name bicomplex.

_{1}, i

_{2}, i

_{3}}, where i

_{3}= i

_{1}i

_{2}= i

_{2}i

_{1}, is the basis of a bicomplex number, which can also be represented in hypercomplex form as a 4D number with real coefficients, as follows:

_{1}

^{2}= +1. A set of 4D commutative hypercomplex algebras is also proposed in [35], for which i

_{2}

^{2}= +1.

## 3. Properties of Commutative Bicomplex Numbers

_{S}and A

_{V}are called the scalar part and vector part of the bicomplex number A, respectively, while a

_{0}, a

_{1}, a

_{2}, and a

_{3}are real numbers. The three imaginary units of the bicomplex number A will be denoted by j, i, and k.

^{2}= −1 is satisfied; i is the vector unit with i

^{2}= −1, while the square of the imaginary unit k is equal to +1. The properties of the imaginary units of an RBQ are as follows [36]:

_{S}and the vector A

_{V}parts of A (5) can be determined by means of the vector-conjugated reduced biquaternion A

^{+}(7), as follows:

_{j}, θ

_{i}, and θ

_{k}take values in the intervals:

_{k}in (15) is explained by the impossibility of applying Euler’s formula, since k is not the classic imaginary unit (k

^{2}= +1) and, therefore, the hyperbolic functions sinh and cosh and Lorentz transformation are used instead of the trigonometric sin and cos.

^{+}and B

^{+}—perform in the following way:

## 4. Outline of the Bicomplex Orthogonal DSP Algorithm Design Procedure

_{1}= 90°, which transforms the pole rotation substitution into an orthogonal complex one.

_{1}and θ

_{2}), and for this reason bicomplex orthogonality may be interpreted in several ways. A possible approach to define bicomplex orthogonality is to assume that each of the two complex numbers constituting the bicomplex number is itself orthogonal, i.e., θ

_{1}= θ

_{2}= 90°. Hence, the following substitution, denoted by R → oBC, where “o” stands for orthogonal, is received:

_{1}= 90°, which provides a zero real part and unity imaginary part. Expanding this logic with regard to a bicomplex number, in order to obtain orthogonality, its scalar part (cis θ

_{1}) must be set to zero, thus reducing the bicomplex number to its vector part (cis θ

_{2}) only. Therefore, bicomplex orthogonality will be achieved via the following transformation:

_{2}, of a bicomplex number being unity. Since the vector part itself is a complex number, the requirement could be its modulus (Euclidean norm) to be equal to one, which is accomplished for different numerical values of θ

_{2}, such as those represented in Table 2.

_{2}being one, while its imaginary part is zero, which can be achieved when θ

_{2}= 0°, or vice-versa (θ

_{2}= 90°) [24]. In this work, it is assumed that θ

_{2}= 45°, resulting in the following orthogonal bicomplex transformation:

_{R}(z), H

_{oC}(z), and H

_{oBC}(z) denote real, complex, and orthogonal in regard to the vector unit i, and bicomplex orthogonal algorithms in the complex z-domain, respectively.

_{2}= 45° are as follows:

_{R}(z) rotate clockwise and counterclockwise at an angle of θ

_{1}= 90°, and double in number. The rotation of the single pole of a bilinear real digital algorithm, resulting in a pair of complex–conjugate poles on the imaginary axis of the corresponding orthogonal complex algorithm, is shown in Figure 2.

_{oBC}(z)—when starting from a real algorithm H

_{R}(z), which is first transformed into an orthogonal complex H

_{oC}(z) algorithm, are shown in Figure 3.

_{oBC}(z) can be represented by its scalar part H

_{oS}(z) and vector part H

_{oV}(z), both of which are complex coefficient functions of the 2N-order. In accordance with bicomplex number arithmetic, the scalar and vector parts can be derived via the bicomplex orthogonal transfer function and its vector conjugate, as follows:

_{oBC}(z) can be also represented by the four real coefficient transfer functions of the 4N-order H

_{o}

_{1}(z), H

_{o}

_{2}(z), H

_{o}

_{3}(z), and H

_{o}

_{4}(z), which comprise the scalar H

_{oS}(z) and vector H

_{oV}(z) parts, but on the other hand can also be determined via them, as follows:

## 5. Bilinear Orthogonal Bicomplex DSP Algorithm Derivation—A Numerical Example

_{R}(z) (32), the following transfer functions describing orthogonal complex and orthogonal bicomplex digital algorithms will be obtained:

_{oC}(z), the transformation (29) will produce the orthogonal bicomplex transfer function H

_{oBC}(z):

_{oS}(z) and H

_{oV}(z), respectively, are:

_{o}

_{1}(z), H

_{o}

_{2}(z), H

_{o}

_{3}(z), and H

_{o}

_{4}(z), are determined as follows, respectively:

_{oBC}(z):

_{oR}

_{1}(z) and H

_{oR}

_{2}(z) (34), respectively:

_{R}(z) (32) is low-pass (LP), H

_{oR}

_{1}(z) and H

_{oR}

_{2}(z) (34) will both be of band-pass (BP) type, while the four bicomplex orthogonal transfer functions—H

_{o}

_{1}(z), H

_{o}

_{2}(z), H

_{o}

_{3}(z), and H

_{o}

_{4}(z) (37)—are also of BP type, but with two passbands each. A high-pass (HP) H

_{R}(z) will produce both BP and band-stop (BS) transfer functions (34), and in the bicomplex case the transfer functions (37) will again be multiband, having two passbands for the BP type and two stopbands for the BS type.

_{R}(z) (32) is 0.99, the real bilinear digital algorithm becomes narrowband and LP; its magnitude and phase responses, together with those of the corresponding orthogonal complex H

_{oR}

_{1}(z) (34) and the bicomplex H

_{o}

_{1}(z) (37) constituents, are shown in Figure 6.

_{R}(z) is c = 1−a, for H

_{oR}

_{1}(z) it is c

_{R}

_{1}= 1−a

^{2}, and for H

_{o}

_{1}(z) it is c

_{o}

_{1}= 1−a

^{4}. Since the experimental results for the other complex and bicomplex orthogonal transfer functions—H

_{oR}

_{2}(z), H

_{o}

_{2}(z), H

_{o}

_{3}(z), and H

_{o}

_{4}(z)—are similar, they are not represented in this work.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- BenYoussef, N.; Bouzid, A. Color edge detection using quaternion convolution and vector gradient. In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), Porto, Portugal, 27 February–1 March 2017; Volume 4, pp. 135–139. [Google Scholar] [CrossRef]
- Ortolani, F.; Comminiello, D.; Uncini, A. The widely linear block quaternion least mean square algorithm for fast computation in 3D audio systems. In Proceedings of the IEEE International Workshop on Machine Learning for Signal Processing (MLSP), Vietri sul Mare, Italy, 13–16 September 2016. [Google Scholar] [CrossRef]
- Gonzales, A.M.; Grigoryan, A.M. Fast Retinex for color image enhancement: Methods and algorithms. In Proceedings of the SPIE 9411, Conference on Mobile Devices and Multimedia: Enabling Technologies, Algorithms, and Applications, San Francisco, CA, USA, 10–11 February 2015. [Google Scholar] [CrossRef]
- Tao, J.W. Performance analysis for interference and noise canceller based on hypercomplex and spatio-temporal-polarisation processes. IET Radar Sonar Navig.
**2013**, 7, 277–286. [Google Scholar] [CrossRef] - Carre, P.; Denis, P.; Fernandez-Maloigne, C. Spatial color image processing using Clifford algebras: Application to color active contour signal image and video processing. Signal Image Video Processing
**2012**, 8, 1357–1372. [Google Scholar] [CrossRef] - Wong, W.K.; Lee, G.C.; Loo, C.K.; Lock, R. Quaternion based fuzzy neural network classifier for MPIK dataset’s view-invariant color face image recognition. Informatica
**2013**, 37, 181–192. [Google Scholar] - Nagase, T.; Komata, M.; Araki, T. Secure signals transmission based on quaternion encryption scheme. In Proceedings of the 18th International Conference on Advanced Information Networking and Applications, (AINA 2004), Fukuoka, Japan, 29–31 March 2004; Volume 2, pp. 35–38. [Google Scholar] [CrossRef]
- Gebre-Egziabher, D.; Elkaim, G.H.; Powell, J.D.; Parkinson, B.W. A gyro-free quaternion-based attitude determination system suitable for implementation using low cost sensors. In Proceedings of the IEEE Position Location and Navigation Symposium, San Diego, CA, USA, 13–16 March 2000; pp. 185–192. [Google Scholar] [CrossRef] [Green Version]
- Olsson, T.; Bengtsson, J.; Robertsson, A.; Johansson, R. Visual position tracking using dual quaternions with hand-eye motion constraints. In Proceedings of the IEEE International Conference on Robotics and Automation, (ICRA’03), Taipei, Taiwan, 14–19 September 2003; Volume 3, pp. 3491–3496. [Google Scholar] [CrossRef] [Green Version]
- Cai, X.; Zhou, W.; Cen, G.; Qiu, W. Saliency detection for stereoscopic 3D images in the quaternion frequency domain. 3D Res.
**2018**, 9, 1–12. [Google Scholar] [CrossRef] - Niknam, S.; Roy, A.; Dhillon, H.S.; Singh, S.; Banerji, R.; Reed, J.H.; Saxena, N.; Yoon, S. Intelligent O-RAN for Beyond 5G and 6G Wireless Networks. arXiv
**2020**, arXiv:2005.08374. [Google Scholar] - De Alwis, C.; Kalla, A.; Pham, Q.V.; Kumar, P.; Dev, K.; Hwang, W.J.; Liyanage, M. Survey on 6G frontiers: Trends, applications, requirements, technologies and future research. IEEE Open J. Commun. Soc.
**2021**, 2, 836–886. [Google Scholar] [CrossRef] - Habibi, M.A.; Nasimi, M.; Han, B.; Schotten, H.D. A comprehensive survey of RAN architectures toward 5G mobile communication system. IEEE Access
**2019**, 7, 70371–70421. [Google Scholar] [CrossRef] - Wang, C.; Renzo, M.D.; Stanczak, S.; Wang, S.; Larsson, E.G. Artificial intelligence enabled wireless networking for 5G and beyond: Recent advances and future challenges. IEEE Wirel. Commun.
**2020**, 27, 16–23. [Google Scholar] [CrossRef] [Green Version] - Helstrom, C.W. Statistical Theory of Signal Detection; Pergamon: New York, NY, USA, 1960; ISBN 9781483156859. [Google Scholar]
- Crystal, T.H.; Ehrman, L. The design and applications of digital filters with complex coefficients. IEEE Trans. Audio Electroacoust.
**1968**, 16, 315–320. [Google Scholar] [CrossRef] - Jantzi, S.A.; Martin, K.W.; Sedra, A.S. The effects of mismatch in complex bandpass ΔΣ modulators. In Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS’96), Atlanta, GA, USA, 12–15 May 1996; Volume 1, pp. 227–230. [Google Scholar] [CrossRef]
- Nikolova, Z.V.; Stoyanov, G.K.; Iliev, G.L.; Poulkov, V.K. Complex coefficient IIR digital filters. In Digital Filters; Márquez, F., Ed.; IntechOpen: London, UK, 2011; pp. 209–239. ISBN 978-953-307-190-9. [Google Scholar] [CrossRef] [Green Version]
- Valkova-Jarvis, Z.V.; Mihaylova, D.A.; Mihovska, A.D.; Iliev, G.L. Adaptive complex filtering for narrowband jamming mitigation in resource-constrained wireless networks, IGI Global. Int. J. Interdiscip. Telecommun. Netw.
**2020**, 12, 46–58. [Google Scholar] [CrossRef] - Li, S.; Leng, J.; Fei, M. The quaternion-Fourier transform and applications. In Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering (LNICST); Springer: Cham, Switzerland, 2019; Volume 262, pp. 157–165. [Google Scholar]
- Osaco, H.; Ueda, K.; Takahashi, S.-I. Digital filter with eight elements hypercomplex coefficient. In Proceedings of the 12th European Conference on Circuit Theory & Design (ECCTD’95), ITU, Istanbul, Turkey, 27–31 August 1995; Volume 1, pp. 659–662. [Google Scholar]
- Toyoshima, H.; Higuchi, S. Design of hypercomplex all-pass filters to realize complex transfer functions. In Proceedings of the 2nd International Conference on Information, Communications and Signal Processing (ICICSP’99), Weihai, China, 28–30 September 1999; pp. 1–5. [Google Scholar]
- Kalinovsky, Y.; Boyarinova, Y.; Khitsko, I.; Oleshchenko, L. Digital filters optimization modelling with non-canonical hypercomplex number systems. In Advances in Intelligent Systems and Computing; Springer: Cham, Switzerland, 2020; Volume 938, pp. 448–458. [Google Scholar] [CrossRef]
- Valkova-Jarvis, Z.V.; Mihaylova, D.A.; Stoynov, V.R. Efficient orthogonal bicomplex bilinear DSP algorithm design. Sci. J. Riga Tech. Univ. Electr. Control. Commun. Eng.
**2020**, 16, 30–36. [Google Scholar] [CrossRef] - Alfsmann, D.; Göckler, H.G. Hypercomplex bark-scale filter bank design based on allpass-phase specifications. In Proceedings of the European Signal Processing Conference (EUSIPCO 2012), Bucharest, Romania, 27–31 August 2012; pp. 519–523. [Google Scholar] [CrossRef]
- Kamata, M.; Takahashi, S.-I. Orthogonal filter with hypercomplex coefficients, including cases of complex and real ones. In Proceedings of the ECCTD’97, Budapest, Hungary, 31 August–3 September 1997; pp. 594–598. [Google Scholar]
- Okuda, M.; Kamata, M.; Takahashi, S.-I. Realization of an orthogonal filter with hypercomplex coefficients. Electron. Commun. Jpn. (Part III: Fundam. Electron. Sci.)
**2002**, 85, 52–60. [Google Scholar] [CrossRef] - Li, Y.N. Quaternion polar harmonic transforms for color images. IEEE Signal Process. Lett.
**2013**, 20, 803–806. [Google Scholar] [CrossRef] - Hu, B.; Zhou, Y.; Li, L.-D.; Zhang, J.-Y.; Pan, J.-S. Polar linear canonical transform in quaternion domain. J. Inf. Hiding Multimed. Signal Processing
**2015**, 6, 1185–1193. [Google Scholar] - Cantor, I.L.; Solodovnikov, A.S. Hyperbolic Numbers; Science: Moscow, Russia, 1973. (In Russian) [Google Scholar]
- Luna-Elizarrarás, M.E.; Shapiro, M.; Struppa, D.C.; Vajiac, A. The bicomplex numbers. In Book Bicomplex Holomorphic Functions (Frontiers in Mathematics), 1st ed.; Birkhäuser: Basel, Switzerland, 2015; ISBN 978-3-319-24866-0. [Google Scholar]
- Cockle, J. On certain functions resembling qaternions and on a new imaginary in algebra. Lond. Dublin Edinb. Philos. Mag.
**1848**, 33, 43–59. [Google Scholar] - Corrado, S. Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici (Real representations of complex forms and hyperalgebraic bodies). Math. Ann.
**1892**, 40, 413–467. [Google Scholar] [CrossRef] - Ell, T.A. Quaternion Fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems. In Proceedings of the 32nd IEEE Conference on Decision Control, San Antonio, TX, USA, 15–17 December 1993; Volume 2, pp. 1830–1841. [Google Scholar] [CrossRef]
- Clyde, C.M.; Davenport, M. A commutative hupercomplex algebra with associated function theory. In Book Clifford Algebra with Numeric and Symbolic Computations; Ablamowicz, R., Ed.; Birkhauser: Boston, MA, USA, 1996; pp. 213–227. [Google Scholar]
- Schutte, H.D.; Wenzel, J. Hypercomplex numbers in digital signal processing. In Proceedings of the IEEE Singapore International Symposium on Circuits and Systems (ISCAS), Windsor, ON, USA, 1–3 May 1990; pp. 1557–1560. [Google Scholar] [CrossRef]
- Luna-Elizarrarás, M.E.; Alpay, D.; Struppa, D.C.; Shapiro, M. Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis; Springer: Cham, Switzerland, 2014. [Google Scholar]
- Watanabe, E.; Nishihara, A. A synthesis of a class of complex digital filters based on circuitry transformations. IEICE Trans.
**1991**, E-74, 3622–3624. [Google Scholar]

**Figure 1.**Possible approaches to design a bicomplex DSP algorithm starting from a real or complex algorithm.

**Figure 2.**Pole rotation of a real bilinear DSP algorithm when applying the transformation R → oC (28), and then pole rotation of a complex bilinear DSP algorithm when applying the transformation oC → oBC (29).

**Figure 3.**Synthesis of the orthogonal bicomplex DSP algorithm by the sequence of transformations (28) and (29): R→oC→oBC.

**Figure 6.**Magnitude and phase responses of the (

**a**) real LP section, (

**b**) orthogonal complex BP realization, and (

**c**) orthogonal bicomplex multiband realization.

Bicomplex Numbers | i_{1} | i_{2} | i_{3}= i_{1}i_{2} = i_{2}i_{1} |
---|---|---|---|

Tessarines | ${i}_{1}^{2}=-1$ | $\begin{array}{l}{i}_{2}^{2}=+1\\ {i}_{2}\ne \pm 1\end{array}$ | ${i}_{3}^{2}=+1$ |

Reduced biquaternions | ${i}_{1}^{2}=-1$ | ${i}_{2}^{2}=-1$ | ${i}_{3}^{2}=+1$ |

Complex hyperbolic numbers | ${i}_{1}^{2}=-1$ | $\begin{array}{l}{i}_{2}^{2}=+1\\ {i}_{2}\ne \pm 1\end{array}$ | ${i}_{3}^{2}=-1$ |

Hyperbolic complex numbers | $\begin{array}{l}{i}_{1}^{2}=+1\\ {i}_{1}\ne \pm 1\end{array}$ | ${i}_{2}^{2}=-1$ | ${i}_{3}^{2}=-1$ |

Hyperbolic dual numbers | $\begin{array}{l}{i}_{1}^{2}=+1\\ {i}_{1}\ne \pm 1\end{array}$ | $\begin{array}{l}{i}_{2}^{2}=0\\ {i}_{2}\ne 0\end{array}$ | $\begin{array}{l}{i}_{2}{i}_{3}=0\\ {i}_{1}{i}_{3}={i}_{2}\end{array}$ |

Dual hyperbolic numbers | $\begin{array}{l}{i}_{1}^{2}=0\\ {i}_{1}\ne 0\end{array}$ | $\begin{array}{l}{i}_{2}^{2}=+1\\ {i}_{2}\ne \pm 1\end{array}$ | $\begin{array}{l}{i}_{2}{i}_{3}={i}_{1}\\ {i}_{1}{i}_{3}=0\end{array}$ |

Hyper-dual numbers | $\begin{array}{l}{i}_{1}^{2}=0\\ {i}_{1}\ne 0\end{array}$ | $\begin{array}{l}{i}_{2}^{2}=0\\ {i}_{2}\ne 0\end{array}$ | $\begin{array}{l}{i}_{3}^{2}=0\\ {i}_{3}\ne 0\end{array}$ |

$R\to oBC:\hspace{1em}{z}^{-1}\to {z}^{-1}\left[i\left(\mathrm{cos}{\mathsf{\theta}}_{2}+j\mathrm{sin}{\mathsf{\theta}}_{2}\right)\right]$ | θ_{2} |

${z}^{-1}\to i{z}^{-1}$ | 0 |

${z}^{-1}\to i{z}^{-1}\frac{\sqrt{2}}{2}\left(1+j\right)$ | π/4 |

${z}^{-1}\to i{z}^{-1}j$ | π/2 |

${z}^{-1}\to -i{z}^{-1}\frac{\sqrt{2}}{2}\left(1+j\right)$ | 3π/4 |

${z}^{-1}\to -i{z}^{-1}$ | π |

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**MDPI and ACS Style**

Valkova-Jarvis, Z.; Poulkov, V.; Stoynov, V.; Mihaylova, D.; Iliev, G.
A Method for the Design of Bicomplex Orthogonal DSP Algorithms for Applications in Intelligent Radio Access Networks. *Symmetry* **2022**, *14*, 613.
https://doi.org/10.3390/sym14030613

**AMA Style**

Valkova-Jarvis Z, Poulkov V, Stoynov V, Mihaylova D, Iliev G.
A Method for the Design of Bicomplex Orthogonal DSP Algorithms for Applications in Intelligent Radio Access Networks. *Symmetry*. 2022; 14(3):613.
https://doi.org/10.3390/sym14030613

**Chicago/Turabian Style**

Valkova-Jarvis, Zlatka, Vladimir Poulkov, Viktor Stoynov, Dimitriya Mihaylova, and Georgi Iliev.
2022. "A Method for the Design of Bicomplex Orthogonal DSP Algorithms for Applications in Intelligent Radio Access Networks" *Symmetry* 14, no. 3: 613.
https://doi.org/10.3390/sym14030613