Cheong Took, Clive

Xia, Yili

Quaternion least-mean-quare (QLMS) algorithm has been thoroughly investigated in the past for the adaptive filtering of 3D and 4D signals. However, QLMS is restricted to one-channel processing of such 3D and 4D processes. To address this shortcoming, we proposed the multichannel QLMS so that we can exploit the additional information from the multitude of 3D-4D vector sensors. For rigour, we provide the stability bound of the multichannel QLMS and demonstrate its robustness against noise. Simulations on real world 4D signals support the analysis

Royal Holloway - Pure

MULTICHANNEL QUATERNION LEAST MEAN SQUARE ALGORITHMClive Cheong Took∗ and Yili Xia†∗Department of Electronic Engineering, Royal Holloway University of London†School of Information Science and Engineering, Southeast University, ChinaEmails: clive.cheongtook@rhul.ac.uk, yili xia@seu.edu.cnABSTRACTQuaternion least-mean-quare (QLMS) algorithm has beenthoroughly investigated in the past for the adaptive filteringof 3D and 4D signals. However, QLMS is restricted to one-channel processing of such 3D and 4D processes. To addressthis shortcoming, we proposed the multichannel QLMS sothat we can exploit the additional information from the mul-titude of 3D-4D vector sensors. For rigour, we provide thestability bound of the multichannel QLMS and demonstrateits robustness against noise. Simulations on real world 4Dsignals support the analysis.Index Terms— hypercomplex, quaternions, adaptive fil-ters, multichannel processing1. INTRODUCTIONThere are many natural phenomena that are inherently three-dimensional (3D) and four-dimensional (4D), yet it is com-mon to treat these phenomena as multichannel signals insteadof 3D or 4D signals. Perhaps due to convention and conve-nience or perhaps due to lack of modelling in 3D and 4D.Learning in the hypercomplex domain enables us us to pro-cess 3D and 4D data as a single entity rather than modellingas a multichannel entity, hence preserving the integrity of thedata. The cross-product a × b is well defined for R3. Thissimple yet useful mathematical operation indicates the suit-ability of quaternions for 3D modelling, since the quaternionproduct involves the cross product asx1x2 = Sx1Sx2 − V x1V x2 + Sx2V x2 (1)+Sx1V x2 + V x1 × V x2︸ ︷︷ ︸cross productwhere S and V denote the scalar and vector part of the quater-nion variable. To this end, quaternion signal processing hasemerged as a candidate tool to cater specifically for these 3Dand 4D processes. For instance, quaternion least mean squarealgorithm (QLMS) has been exploited in wind forecastingand 3D audio systems [1], quaternion principal componentanalysis for colour images [2], and weighted linear combinerfor modelling 3D tremors [3].The emergence of quaternion algorithms and theories hasparticularly sparked an interest in the adaptive signal pro-cessing community. Nascimento et al. have proposed lowcomplexity adaptive filters [4, 5], and Liu proposed noveladaptive beamforming algorithms [6]. The analysis of thesequaternion-valued adaptive filters have been investigated in[7], whereas Mars et al. discussed the derivations of quater-nion adaptive filters [8]. However, the adaptive filtering ofmultiple 3D and 4D processes simultaneously is still lackingand is addressed in this work in the form of multichannelQLMS.The remainder of this paper is organised as follows. Fun-damentals of quaternions are revisited in Section 2. Themultichannel QLMS is derived in Section 3 followed bysome remarks regarding issues with the implementation ofmultichannel QLMS. Section 4 provides simulation resultsdemonstrating the performance of the proposed algorithmand finally, Section 5 concludes this paper.2. QUATERNIONS AND THEINVOLUTION-GRADIENTA quaternion x = xa+ ıxb+ xc+κxd is comprised of a real(scalar) part xa and three imaginary components xb, xc, xd inits vector part. The vector product of these three imaginaryparts makes the quaternion product between x and y noncom-mutative, i.e.xy 6= yx (2)As in the complex case, the conjugate of a quaternion variableis obtained by negating the imaginary parts asx = xa − ıxb − xc − κxd (3)The additional degrees of freedom from complex to quater-nion domain means that there exist other useful algebraic op-erations such as the quaternion involutions (self-inverse map-pings)xη = −ηxη, η ∈ {ı, , κ} (4)Alternatively, these involutions can be expressed asxı = xa + ıxb − xc − κxdx = xa − ıxb + xc − κxdxκ = xa − ıxb − xc + κxd (5)In this way, these involutions enable componentwise con-jugation. For example, xı∗ means the conjugation of onlythe ı−imaginary component of x. The relationships betweenthese involutions are governed byx∗ =12(xı + x + xκ − x)x =12(xı∗ + x∗ + xκ∗ − x∗) (6)The pair of relationships in (6) will be used in the derivationof the multichannel QLMS algorithms.2.1. The Involution GradientThe so called pseudogradient is often employed to calculategradients due to its straightforward formulation as the sum ofcomponentwise gradients, i.e.∂J∂x=∂J∂x+ ı∂J∂xı+ ∂J∂x+ κ∂J∂xκ(7)However, the process of calculating componentwise gradi-ents is tedious. To address this issue, we introduced the HR-calculus [9], so that derivatives can be obtained directly in thequaternion space. For example, (∂x∗)/(∂x∗) = 1, whereasbased on (6) we have∂x∗∂x=∂[0.5(xı + x + xκ − x)]∂x= −0.5x (8)Due to the relationships in (6), we can also take quaterniongradients with respect to the involutions (5), which was re-ferred as the involution-gradient, i.e.∇xηJ =∑η={ı,,κ}∂J∂xη(9)It was shown in [10] that the steepness of the involution-gradient was marginally greater than the HR∗−derivative, yetboth lead to the same solution at the local minima. It is theinvolution-gradient that we will adopt to derive multichannelQLMS algorithms.3. THE MULTICHANNEL QLMS ALGORITHMFor the multichannel QLMS algorithm, the qth output can beexpressed in terms of its inputs x and filter coefficients w asyq(n) =P∑p=1wHpqxp(n) q = 1, . . . , Q (10)wherewpq = [wpq,1 wpq,2 · · · wpq,L]Txp(n) = [x(n) x(n− 1) · · · x(n− L+ 1)]TThe superscripts (·)H and (·)T denote the Hermitian and trans-pose operator respectively. More generally, the output vectory(n) can be expressed asy(n) = WHx(n)y1(n)y2(n)...yQ(n) =w11 w12 . . . w1Qw21 w21 . . . w2Q............wP1 wP2 . . . wPQH x1(n)x2(n)...xP (n)(11)The error for the qth channel can be expressed aseq(n) = dq(n)− yq(n) q = 1, ..., Q (12)The cost function of qth channel can be formulated as [11]Jq(n) = eq(n)e∗q(n) (13)For the qth channel, the update for its coefficients can be com-puted using the steepest descent:Wq(n+ 1) = Wq(n)− µ∇Jq(n) (14)where Wq is given byw1q,1 w1q,2 . . . w1q,Lw2q,1 w2q,2 . . . w2q,L............wPq,1 wPq,2 . . . wPq,LThe derivative in (14) can be formulated based on the involu-tion gradient as∇Jq(n) =∑η={ı,,κ}eq(n)∂e∗q(n)∂Wηq (n)+∂eq(n)∂Wηq (n)e∗q(n) (15)As the conjugate of the error (12) is given bye∗q(n) = d∗q(n)−P∑p=1xHp (n)wpq q = 1, ..., Q (16)the involution relationship (6) means that ∂e∗q(n)∂Wηq (n)= 0,since Eq. (16) is dependent of Wη∗q (n), but independentof Wηq (n). Similarly, the other derivative in (15) can becalculated as∂eq(n)∂Wηq (n)= −0.5x(n) (17)Based (14)-(15), the coefficients corresponding to the qthchannel output of multichannel QLMS can be learnt asWq(n+ 1) = Wq(n) + µX(n)e∗q(n) (18)Samples0 10 20 30 40 50 60 70 80 90 100MSE [dB]10-210-1100101102SNR= -40 dBSNR= -30 dBSNR= -20 dBSNR= -10 dBSNR= 0 dB(a)Learning Rate×10-30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Prediction Gain [dB]-300-250-200-150-100-50050multi QLMSQLMS(b)Fig. 1. The left plot (a) shows the learning curves of QLMS (‘-’) and multichannel QLMS(‘-*’). The right plot (b) shows thestability range of the learning rate or stepsize.where any constant has been absorbed in the learning rate/stepsize and X(n) is given byx1(n) x1(n− 1) . . . x1(n− L+ 1)x2(n) x2(n− 1) . . . x2(n− L+ 1)............xP (n) xP (n− 1) . . . xP (n− L+ 1)Instead of updating each qth adaptive filter individually (18),we can generalise the update of the multichannel QLMS asW(n+ 1) = W(n) + µx(n)eH(n) (19)where the variables W(n) and x(n) have the same dimen-sionality as in (11), and the error vector is given bye(n) = [e1(n) e2(n) · · · eq(n)]TEach learning coefficient of (19) can be expressed aswpq,l(n+ 1) = wpq,l(n) + ∆wpq,l(n+ 1) (20)= wpq,l(n) + µxp(n− l)e∗q(n) l = 1, ..., LRemark 1: At first sight from (20), it looks like each updateof the qth adaptive filter does not depend on the other filters,since its update depends only on the qth error. However, onexpanding the qth error, we have∆wpq,l(n+1) = µxp(n)[d∗q(n)−P∑i=1xHi (n)wiq(n)](21)Notice that each learning coefficient is updated based on allinput channels {xp}Pp=1. This leads to the ‘dependency’ be-tween each channel adaptive filter, as observed in [11].Remark 2: Remark 1 also demonstrates that the multichannelQLMS exploits cross-channel correlation between 3D or 4Dprocesses, since the update depends on the correlation termE{xp(n)∑Pi=1 xi(n)} upon taking its expectation.Remark 3: The correlation term E{xp(n)∑Pi=1 xi}, how-ever, does not guarantee that the coupling within each quater-nion input channel xi(n) is exploited. To do so, we cantake advantage of advances in quaternion statistics by aug-menting the input vector with the three involutions, i.e.xa(n) = [x xı x xκ] [12].Remark 4: The blockbase implementation in (19) is fasterthan channelwise implementation in (18), if implemented inMatlab. The latter requires loop-based coding, whereas theformer can be computed much faster due to the ‘vectorisa-tion’ facility implemented in Matlab.Remark 5: It is straighforward to show that the range ofvalues of the learning rate for multichannel QLMS (19) is0 < µ < 2λmax, where λmax is maximum eigenvalue of themultichannel correlation matrix E{xxH}. A more practicalbound of λmax is LN∑Ni=0∑Pp=1 x2p(n − i), where N is thenumber of samples that has been processed, and L is the filterlength.4. SIMULATIONThe performance and convergence properties of our pro-posed multichannel QLMS was assessed against QLMS in aforecasting application with a prediction horizon of one stepahead. For this forecasting task, wind data which comprisedof two anemometers (sensors) recordings of wind speeds(East-West, North-South, Vertical direction) and temperature(degree Celsius) was considered. Hence, the total numberof channels for quaternion-valued processes were two. Forrigour, two performance metrics were used, i.e. the traditionalmean-square error (MSE) and the prediction gain Rp, whichcan computed asRp = 10 log10(σ2x/σ2n) [dB]where σ2x and σ2n are the estimated sample variances of theinput and the additive white Gaussian noise of all channels.To assess the robustness of the algorithms against noise, 50Monte Carlo simulations were run and averaged out for eachvalue of signal-noise-ratio (SNR) and shown in the left plotof Fig. 1. As expected, when the SNR was decreased, theperformance of multichannel QLMS (denoted as ‘-*’) andQLMS (denoted as ‘-’) deteriorated.Remark 6: The cross-correlation between the two quaternion-valued channels benefited the multi-channel QLMS and madeit more robust against noise, whereas the QLMS could notexploit this correlation due to its inherent uni-channel pro-cessing. This was expected due to Remark 2.To investigate the convergence and stability properties ofQLMS and multichannel QLMS, an experiment was carriedout to measure the prediction gain of both algorithms over arange of stepsizes/learning rates, and the results are shown inthe right plot of Fig. 1. If we consider the performance ofboth algorithms to be ‘reasonable’ when the prediction gainis greater than 0 dB, then the range of stepsize for QLMS is0 < µ < 0.68× 10−3whereas for the multichannel QLMS, the range is0 < µ < 0.35× 10−3Remark 7: In our simulation, the range of values of step-size for QLMS is approximately twice that of multichannelQLMS. This observation confirms Remark 5 (i.e. 0 < µ <2λmax), since the two channels of multichannel QLMS makes itlikely that its λmax is twice that of the one channel of QLMS.5. CONCLUSIONWe have proposed the multichannel QLMS algorithm that canprocess 3D and 4D processes simultaneously. It was shownthat the additional information of the multichannel processingover the unichannel processing made our proposed algorithmmore robust against noise. However, the cost of processingmore channels means that the stability range of the multi-channel QLMS is likely to be much less than that of QLMS.This is due to the dependency between channels, i.e. the er-ror of one channel can affect the other channels as well, fol-lowing Remark 2. Future work will involve the investigationof augmented statistics in deriving the multichannel QLMS,however, we expect the same results as in this paper, whencomparing it against the widely linear QLMS.6. REFERENCES[1] F. Ortolani, D. Comminiello, and A. Uncini, “Thewidely linear block quaternion least mean square algo-rithm for fast computation in 3d audio systems,” IEEEInternational Workshop on Machine Learning for SignalProcessing, Sept. 2016.[2] N. Le Bihan and S. J. Sangwine, “Quaternion principalcomponent analysis of color images,” IEEE Interna-tional Conference on Image Processing, Sept. 2003.[3] K. Adhikari, S. Tatinati, W. T. Ang, K. C. Veluvolu,and K. Nazarpour, “A quaternion weighted Fourier lin-ear combiner for modeling physiological tremor,” IEEETransactions on Biomedical Engineering, vol. 63, no.11, 2016.[4] F. G. A. Neto and V. H. Nascimento, “A novel reduced-complexity widely linear QLMS algorithm,” IEEE Sta-tistical Signal Processing Workshop, June 2011.[5] Vı´tor H. Nascimento and Rodrigo C. de Lamare, “Alow-complexity strategy for speeding up the conver-gence of convex combinations of adaptive filters,” IEEEInternational Conference on Acoustics, Speech, and Sig-nal Processing, March 2012.[6] X. Lan and W. Liu, “Fully quaternion-valued adaptivebeamforming based on crossed-dipole arrays,” Electron-ics, vol. 6, no. 2, pp. 34, 2017.[7] Min Xiang, Sithan Kanna, and Danilo P. Mandic, “Per-formance analysis of quaternion-valued adaptive filtersin nonstationary environments,” IEEE Transactions onSignal Processing, vol. 66, no. 6, pp. 1566–1579, 2018.[8] Q. Barthelemy, A. La Rue, and J. Mars, “About QLMSderivations,” IEEE Signal Processing Letters, vol. 21,no. 2, pp. 240–243, 2014.[9] D. Mandic, C. Jahanchahi, and C. C. Took, “A quater-nion gradient operator and its applications,” IEEE Sig-nal Processing Letters, vol. 18, no. 1, pp. 47–50, 2011.[10] C. Jahanchahi, C. C. Took, and D. Mandic, “On gradi-ent calculation in quaternion adaptive filtering,” IEEEInternational Conference on Acoustics, Speech and Sig-nal Processing, pp. 3773–3776, 2012.[11] J. Benesty, T. Gansler, Y. Huang, and M. Rupp, “Adap-tive algorithms for MIMO acoustic echo cancellation,”in Audio Signal Processing for Next-Generation Multi-media Communication Systems, Y. Huang and J. Ben-esty, Eds., pp. 119–147. Springer, 2007.[12] C. Cheong Took and D. Mandic, “Augmented second-order statistics of quaternion random signals,” vol. 2,pp. 214–224, 2011.

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Multichannel Quaternion Least Mean Square Algorithm

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