15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP HYPERCOMPLEX ALGEBRAS IN DIGITAL SIGNAL PROCESSING: BENEFITS AND DRAWBACKS Daniel Alfsmann∗, Heinz G. Göckler∗, Stephen J. Sangwine†, and Todd A. Ell‡ ∗Digital Signal Processing Group †Dept. Electron. Syst. Engineering ‡Goodrich Sensor Systems, Ruhr-Universität Bochum University of Essex, Wivenhoe Park Burnsville, MN 55306, USA 44780 Bochum, Germany Colchester CO4 3SQ, UK email: [email protected] email: [email protected] email: [email protected] ABSTRACT R−,C−,H−algebras and the hyperbolic algebras being investi- This tutorial contribution presents a short historical introduction gated in more detail only most recently [3, 4, 5]. and a survey of hypercomplex algebras in conjunction with some It is a commonplace in mathematics that some or all of the fol- beneficial applications in predominantly time-based digital signal lowing fundamental properties of R and C may vanish in hyper- processing. Potential advantages and shortcomings of hypercom- complex algebras of higher dimension: Commutativity, associativ- plex digital signal processing are discussed. ity and multiplicative inverse. If an algebra is not a division alge- bra, some elements may lack a multiplicative inverse and divisors 1. INTRODUCTION of zero exist with the consequence that the product of two non-zero numbers may vanish . This was thoroughly investigated by WEIER- For a long time, mathematicians have studied various (hypercom- STRASS (1815-97) [6]. plex) algebras as a self-contained discipline. Most recently, physi- cists and engineers have gradually grasped the underlying theories This tutorial presents basic material for the understanding of for a variety of applications in signal and image processing, sig- time-based digital signal processing applying hypercomplex alge- nal and system representation and analysis, computer graphics, etc. bra. To this end, we recall the mainstream use of established CDSP under C (section 2). In section 3, we first give our reasons for tran- This tutorial paper presents an introduction to hypercomplex alge- C bras needed for hypercomplex digital signal processing (HCDSP), scending to higher dimension or other algebras, we present the and discusses the potentials and weaknesses of HCDSP. Note that necessary definitions and representations of hypercomplex algebras pure geometric signal processing is beyond the scope of this paper. for HCDSP and, finally, discuss benefits and drawbacks of HCDSP A basic scientific motivation for the investigation of hyper- in view of a variety of suitable applications. In conclusion, in sec- complex algebras is to extend complex (digital) signal processing tion 4 open issues are presented for future research. (CDSP) to HCDSP. We start our presentation with a short histori- cal survey of the advent of complex numbers and the discovery of 2. ESTABLISHED USE OF COMPLEX ALGEBRA IN hypercomplex algebras referring to [1]. SIGNAL PROCESSING (CDSP) Already during the Renaissance it had been recognised in Italy that the real algebra (R) is algebraically not closed under exponen- Almost all physical signals and (equidistantly) sampled versions tiation, when CARDANO (1501-76) and his competitor FONTANA- thereof are real-valued: s(t),s(k) ∈ R. Nevertheless, a widely used TARTAGLIA (1499-1557) were looking for a general solution of 3rd family of versatile and illustrative integral transforms, LAPLACE-, order equations, nowadays known as Cardanic formulae. To over- z-Transform and some varieties of FOURIER Transform (FT) [7], come this limitation, CARDANO in conjunction with BOMBELLI provide a one-to-one mapping to a physically supported complex (1526-72) first introduced complex numbers (C). GAUSS (1777- frequency domain under C. For instance, the FT discloses the spec- 1855) proposed the illustrative complex plane to represent complex tral content of a signal: ULER numbers in Cartesian and polar coordinates, respectively. E FT (1707-83) contributed a great multitude of widely used complex- s(t), s(k) ∈ R ←→ S( jω), S(e jΩ) ∈ C, (1) valued functional relationships. Moreover, GAUSS proved the fun- where any signal is likewise uniquely represented by its spectrum damental theorem of algebra encompassing the result that the com- jΩ plex algebra (C) is algebraically closed. along the frequency “axis” s := jω = j2π f or z := e , Ω = Another track leads to integral or functional transforms, as in- 2π f / fS, respectively (sampling rate: fS = 1/T ). troduced by LAPLACE (1749-1827), FOURIER (1768-1830) and By interchanging time and frequency in (1), time-frequency du- LAURENT (1813-54), respectively: Real or complex functions of ality [7] suggests the existence of complex signals that possess real one or more independent parameters (time, location, etc.) are (in general complex) spectra. The analytic signal, the most impor- mapped onto a complex variable domain, the frequency or spectral tant class of complex signals, has found a variety of applications domain. It is commonplace that a signal spectrum typically gives predominantly in CDSP: i) Efficient digital baseband processing of much more insight into the nature and properties of a signal than narrow-band bandpass signals with reduced sampling rate [8] (e.g. the original signal. homodyne transceiver and single sideband amplitude modulation), A first step beyond complex algebra (C) was made by HAMIL- ii) spectrally compacted data transmission by combining sequential TON (1805-65) discovering the four-dimensional (4-D) quaternions binary data to higher level complex symbols (Quadrature Ampli- (H) in 1843. Soon after HAMILTON’s publication of the quater- tude Modulation, Orthogonal Frequency Division Multiplex, etc.), nion algebra, his student GRAVES and later CAYLEY (1821-95) iii) most efficient processing of discrete orthogonal transforms, such introduced a first kind of 8-D hypercomplex algebras, the Oc- as DFT and FFT, iv) twofold system parallelisation for sample rate taves/Octonions (O) or Cayley numbers, respectively. Moreover, reduction by two, and v) efficient baseband simulation techniques HAMILTON himself introduced still another 8-D hypercomplex al- based on the complex envelope of the analytic bandpass signal. gebra [2], known as complexified quaternions or biquaternions. A The imaginary part of an analytic signal is given by the kind of generalisation to the n-D case was presented by CLIFFORD HILBERT Transform (HT) of its real part. In general, the HT is (1845-79) emphasising, however, geometrical viewpoints and n- implemented by means of a linear and time-invariant (LTI) digital D rotation. Nevertheless, CLIFFORD algebras include the former system [8], for instance, as an FIR system with (zero-phase) im- ©2007 EURASIP 1322 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP pulse response hHT(k) = 2/(kπ) for odd k and 0 elsewhere: whereas multiplication is generally distributive over addition. Obvi- ously, a hypercomplex addition (7) consists of n K-valued additions. FT jΩ jΩ 2 sˆ(k) = hHT(k) ∗s(k) ←→ Sˆ(e ) = − jsgn(sinΩ) · S(e ), (2) A hypercomplex multiplication demands n K-valued multiplica- where (∗) denotes convolution, yielding the digital analytic signal: tions. For instance, the multiplication of two complex numbers, z1 = x1 + iy1, z2 = x2 + iy2,(C is considered as a 2-D hypercom- FT jΩ jΩ plex algebra with K = R and i2 = i2 = −1) according to: s+(k) = s(k) + jsˆ(k) ←→ S+(e ) = S(e )[1+ sgn(sinΩ)]. (3) 2 As a result, the overall spectral bandwidth of any analytic signal z1z2 = x1x2 − y1y2 + i(x1y2 + x2y1), (8) s (k) comprises just half the width of the original real signal s(k), + requires n = 2 real additions and n2 = 4 real multiplications. Of allowing for most of the aforementioned applications exploiting the course, these operations have to be considered when evaluating the potential of unconstrained frequency shifting of analytic signals. computational load of HCDSP. Every associative algebra can be represented by an isomorphic 3. HYPERCOMPLEX ALGEBRAS IN DIGITAL SIGNAL K-valued n× n matrix algebra. For instance, the matrix PROCESSING (HCDSP) q1 −q2 −q3 q4 3.1 Motivation q q −q −q Q = 2 1 4 3 ∈ R4×4 (9) Exploiting the potential of CDSP to advantage, as outlined in sec- q3 q4 q1 q2 tion 2, the following questions related to HCDSP are motivated: −q4 q3 −q2 q1 What are the benefits of HCDSP w.r.t. the above collection of ap- is completely equivalent to the quaternion (5), and all operations plications and beyond? What is the impact on HCDSP, if the un- and properties can likewise be validated with both representations derlying hypercomplex algebra lacks commutativity, associativity, (e.g. that Q is not commutative). However, (9) is highly redundant divisors of zero and/or contains ? Hence, at least the following ba- and therefore computationally inefficient compared to the direct cal- sic system properties, well understood for real and complex DSP culation derived from the algebra’s multiplication table (6). [8], call for thorough investigation in the case of HCDSP: i) LTI An LTI system based on a hypercomplex algebra can always property tightly related to convolution, ii) existence of hypercom- be decomposed either into K-valued basic operations, as in (8), or plex spectral transforms [9], similar to z- and FOURIER Transform, into K-valued subsystems. For the latter, a MIMO (Multiple In- their impact on convolution theorem and the availability of some put Multiple