Bauwelinck, Johan

Demeester, Piet

Li, Haolin

Torfs, Guy

Van Kerrebrouck, Joris

Verbist, Jochem

English

Joris Van Kerrebrouck

Haolin Li

Jochem Verbist

Guy Torfs

Johan Bauwelinck

Piet Demeester

Crossref

Nonlinear MIMO impulse responses determination using pseudo random sequences, and nonlinear compensation

Ghent University Academic Bibliography

Nonlinear MIMO Impulse Responses DeterminationUsing Pseudo Random Sequences, and NonlinearCompensationJoris Van Kerrebrouck, Haolin Li, Jochem Verbist, Guy Torfs, Johan Bauwelinck and Piet DemeesterDepartment of Information Technology (INTEC) – IDLabUniversity of Ghent – imecTechnologiepark 15, B-9052 Zwijnaarde, BelgiumEmail: joris.vankerrebrouck@ugent.beAbstract—This paper shows how to estimate an extendedpolynomial matrix description of a non-linear MIMO system.Both linear and non-linear impulse responses are obtainedby transmitting pseudo random binary sequence (PRBS) test-patterns and can be easily separated thanks to the shift-and-addproperty of PRBS. Further on, a method to equalize such non-linear systems based on this extended polynomial matrix andthe result is compared with a linear zero-forcing equalizer fordifferent channel impairments and in a VPI simulation. Thisproves the presented approach is suited to cancel self-phasemodulation in coherent optical communication systems.I. INTRODUCTIONCommunication channels are pushed to maximize the datathroughput, which is usually limited by the finite frequencyspectrum of these mediums. To overcome this, new communi-cation systems tend towards modulation schemes with higherspectral efficiency and/or stretch the channel bandwidth withequalizers. Therefore, channel estimation and compensationare of crucial importance. Not only are modern communicationchannels browbeaten to transfer more data, multiple channelsare created to establish multiple data streams. Polarizationdivision multiplexing (PDM) and space-division multiplexing(SDM) technologies are used in multiple-input multiple-output(MIMO) systems to achieve multiple parallel data streams tosustain the capacity growth [1]. Along with linear intersymbolinterference (ISI) and co-channel interference (CCI), nonlinearmixing effects are also present, making the impulse responsesof MIMO systems with nonlinear effects complicated. Forexample, the capacity of optical MIMO channels is rapidlyapproaching the capacity limit imposed by these nonlinearfiber effects [2] [3].Channel estimation of MIMO systems is already widelyused and numerous algorithms are available [4], [5]. Typically,the estimation is described using polynomial matrices. Toimprove a MIMO system, the orthogonal eigenvectors of thepolynomial matrix can be used to achieve multiple virtualindependent channels with no crosstalk. To do so, polynomialmatrix singular value decomposition (PMSVD) can be used todecompose a MIMO channel matrix, which can then be usedto recover the transmitted signals corrupted by the channelinterference (CI) at the receiver [6]. However, PMSVD iscomputationally heavy and will only result in a linear com-pensation.To compensate nonlinear effects, nonlinear impulse re-sponses, described with Volterra or power series should becanceled. However, identification of the coefficients is com-putationally burdensome.In this paper, Pseudo Random Bit Sequence (PRBS), alsocalled binary maximum length sequences (MLS) have beenused as excitation to estimate, by cross correlation, the linearimpulse responses of a MIMO channel. These impulse re-sponses are used to build the polynomial matrix. However, thecross correlation of the PRBS stream also reveals additionalimpulse responses that can be attributed to nonlinear channeleffects and that can be used to extend the polynomial matrix.This results in a more accurate channel model. Using thisextended polynomial matrix, a heuristic method is presentedto equalize a nonlinear MIMO system.The structure of this paper is as follows: section II in-troduces an extended version of a polynomial matrix whichincludes extra elements relating to the non-linear impulseresponse. Furthermore, it is shown how all the elements of thismatrix can be estimated based on the transmission of pseudorandom bit sequences. Two important properties of thesesequences allow to extract the different contributions from thereceived signals. In section III, a method to equalize a non-linear MIMO system based on the extended polynomial matrixis proposed and to verify this method, different non-linearcommunication links are compensated in sections IV and Vusing the proposed non-linear equalizer. Finally, conclusionsare drawn in section VI.II. POLYNOMIAL MATRIX REPRESENATION OF A MIMOCHANNELGiven a MIMO link with nT (number of transmitters) inputsand nR (number of receivers) outputs, a nT ×nR polynomialmatrix C(Z) can be used to describe the channel. Fig. 1shows an example of a 2 × 2 MIMO system. Each elementof the matrix C(Z) is a polynomial in the Z-domain. In thispaper the Z-polynomials are translated into vector quantities tosimplify the mathematical processing. The polynomial matrixcan easily be extended to take non-linear terms into account.For example, Eq. (1) shows a polynomial matrix of a 2 × 2MIMO system but extended with the product of the two inputs.[y1y2]=[C11 C12 C13C21 C22 C23]∗ x1x2x1x2 (1)The mathematical operator ∗ represents the convolution.Eq. (1).MIMO Channel C y1x1x2 y2CFig. 1: Representation of a polynomial matrix in a MIMOsystem.A. Obtaining Impulse Response using a PRBSTo determine the different impulse responses constitutingthe channel matrix C(Z), a pseudo random binary sequence(PRBS) is used. A PRBS bit stream is formed using a linearfeedback shift register of length m, given a proper feedbackpolynomial, a maximum length sequence, with period N =2m − 1 can be obtained. If the PRBS is mapped to non-return-to-zero (NRZ) {−1,+1} symbols, the autocorrelationis maximal and results in N when the streams are alignedand equals -1 elsewhere (see Eq. (2)). For creating a perfectKronecker delta with an autocorrelation function, we referto [7]. This makes it a convenient stimulus to calculate theimpulse response of a channel. When a PRBS (u(n)) is sentthrough a channel and the cross correlation of the receivedPRBS (uch(n)) with an undistorted PRBS reveals the channelimpulse response hch(n). This is summarized in Eq. (3)-(5)Ruu(l) =N∑n=1u(n)u(n− l) ={N l = 0−1 l 6= 0(2)uch(n) =∞∑l=−∞hch(l)u(n− l) (3)hch(l) =N∑n=1uch(n)u(n− l) (4)= uch(n) ∗ u(−n) (5)B. PRBS’ Shift and Add PropertyA PRBS sequence has the property that the modulo-2summation or ‘xor’ operation (denoted with ⊕) of a PRBSwith a time-shifted copy of itself yields again the same PRBSbut with a different time-shift. This is commonly known asthe shift-and-add property [8] and is mathematically expressedin Eq. (6). If non-return-to-zero (NRZ) {−1,+1} symbols areused, a modulo-2 sum of two symbols equals the inverse of theproduct of the two symbols (see Eq. (8)). Hence, multiplyingtwo time-shifted versions of an NRZ modulated PRBS resultsin the same PRBS but with yet another time-shift. The time-shift or delay of the resulting PRBS depends only on the time-shift of the copy with respect to the original PRBS. Therefore,we introduce a new function φ(.) in Eq. (7) to indicate thedelay of the resulting PRBS with respect to the original PRBS.This φ(.) function can be determined using Eq. (9), where weuse a discrete logarithm in the finite Galois field of 2m. Thisis the same finite field used to construct the PRBS.u(l) = u(n)⊕ u(n− k) (6)u(n− φ(k)) = u(n)⊕ u(n− k) (7)= −u(n)u(n− k) (8)φ(k) = logGF2m(1 + xk) (9)This property can be extended for three or even more PRBSsequences, which is summarized in Eq. (10) and Eq. (11).u(n1)u(n1 + n2)u(n1 + n2 + n3) = u(n1 + φ(n2, n3))(10)φ(n2, n3) = logGF2m(1 + xn2 + xn2+n3) (11)C. Estimating the non-linear polynomial channel matrix of aMIMO systemUsing the autocorrelation and the sum-and-add properties ofa PRBS, all the elements of the channel matrix represented inEq. (1) can be calculated. To do so, a PRBS stream u(n) anda time-shifted version u(n− k) are applied at the input of thechannel (x1 and x2). Thanks to the shift-and-add property ofthe PRBS, the product of the two inputs yields a third PRBSu(n− φ(k)). This is summarized in Eq. (12)[y1y2]=[h11 h12 h13h21 h22 h23]∗ u(n)u(n− k)u(n− φ(k)) (12)A cross correlation with the original PRBS signals allowsto estimate the different impulse responses. For example,calculating the cross correlation between u(n) and y2 willresult in:(y2 ∗ u(−n))(l) = h21(l) + h22(l − k) + h23(l − φ(k))(13)Hence, this results in the sum of the three impulse responses,but each centered at a different sample moment. If the temporalresolution of the PRBS stream is sufficient, the differentimpulse responses can be easily extracted.III. EQUALIZING IMPULSE RESPONSEThe inverse channel impulse response can be used as azero forcing equalizer. Thus, cascading the channel impulseresponse with the inverse channel impulse response resultsinto a Kronecker Delta δ(n) impulse response (expressed inEq. (14)).δ(n) = hequalizer(n) ∗ hchannel(n) (14)This equalizing polynomial matrix can be constructed usingthe following method. Fig. 2a shows a nonlinear 2 × 2MIMO system. From this, a simplified channel model with theextra non-linear terms is constructed (Fig. 2b), following theestimation method highlighted in section II-B. To equalize thechannel, the channel matrix needs to be inverted. However,this is not possible due to our construction process, as thechannel matrix is not square (2 × 3). To alleviate this, thematrix is extended and a virtual output signal equal to thebest linear approximation of the product of y1 and y2 (denotedỹ12) is added. Fig. 2c shows this. The mathematical derivationEq. (15) shows the terms that contribute to the product y1y2.Other terms of y1y2 which do not have a x1, x2 or x12 areneglected. Notice that x2 = 1 because x is a binary NRZ value{−1,+1}.The extended channel model is given by:y1y2 = (h11 ∗ x1 + h12 ∗ x2 + h13 ∗ x12)︸ ︷︷ ︸y1× (h21 ∗ x1 + h22 ∗ x2 + h23 ∗ x12)︸ ︷︷ ︸y2(15)MIMO Channel y1x1x2 y2(a) 2x2 MIMO channel.ChannelModel y1x1x2 y2x1 x2MatrixHchannel(b) Channel model with theextra non-linear term.y1x1x2 y2x1 x2x1x2Equalizer MatrixHchannel extendMatrixHchannel extend-1y1  y2Channely12(c) 2x2 MIMO channel with equalizer.Fig. 2: Graphical representation of the MIMO equalizationalgorithm. y1y2ỹ12 = h11 h12 h13h21 h22 h23h12h23 + h13h22 h11h23 + h13h21 h11h22 + h12h21︸ ︷︷ ︸hch extend∗ x1x2x1x2(16)To equalize the channel the inverse of the extended channelheq is calculated. Not that y12 represent the product of twooutput signals. x̂1x̂2 = heq y1y2y12The inverted polynomial matrix is computed in the frequencydomain given Eq. (17).F (heq11) F (heq12) F (heq13)F (heq21) F (heq22) F (heq23)F (heq31) F (heq32) F (heq33) =F (hideal) 0 00 F (hideal) 00 0 F (hideal) F (h11) F (h12) F (h13)F (h21) F (h22) F (h23)F (hch extend31) F (hch extend32) F (hch extend33)(17)IV. SIMULATIONThe proposed equalizer is verified in Matlab. Different im-paired 2×2 MIMO channels are constructed and are estimatedusing a PRBS9 with feedback polynomial x9 + x5 + 1. Adelay of 110 symbols is chosen between the PRBSs of thetwo inputs. The product of the PRBS and the delayed PRBSat the input of the system creates a third PRBS with a delay of316 symbols relative to the first PRBS stream φ(110) = 316with φ(.) defined in Eq. (9). The outputs of the MIMO systemare cross-correlated with the perfect PRBS, to attain the rawimpulse response and a window function of 100 symbolslong is used as mask to filter out the specific elements ofthe extended channel matrix shown in Eq. (16). The inversepolynomial matrix is calculated with Eq. (17) and is usedto convolve with the received signal. In the simulations, tworeal inputs are assumed, which can easily be displayed usingconstellation points in a two-dimensional figure. Of course, itis clear that all proposed techniques can be easily extendedtowards higher dimensions.In Table I the different simulated channel responses areshown. The first column shows the received constellationpoints, the second column shown the constellation points aftera linear zero-forcing equalizer and the third column showsthe constellation points after our proposed equalizer. It shouldbe noted that random data transmitted to construct theseconstellation diagrams and not the PRBS9 sequence used to es-timate the impulse responses. The simulation covers the mostcommon MIMO channel distortions. The first distortion shownis a rotation, it can appear in single carrier modulation wherebythe QPSK constellation rotates depending on the receiver localoscillator (LO) phase offset. This linear effect can be easilycompensated with a rotation matrix. The second distortioncomes from IQ mismatch. The third distortions are reflectionsthat cause inter-symbol interference (ISI), this can also becompensated with a linear equalizer. A Trapezoidal constella-tion (Non-linear I) is only possible with a non-linear channel,the linear equalizer can not compensate the trapezoidal shape,which is visible on the constellation diagram of the signal afterthe linear equalizer. The fourth simulation transmits QPSKsymbols with an offset {0,0},{0,1},{1,0},{1,1} modulated ona single carrier. The channel is strongly saturated (Non-linearII), so that the {1,1} symbol is compressed and moved to theorigin. Only the proposed non-linear equalizer can place the4 symbol on the correct coordinates. In the last simulationan asymmetrical PAM4 eye is shown, this effect can occurwhen sending PAM4 symbols through an optical link. Herewe use 2 independent PRBS streams to create the PAM4symbols, one PRBS stream represent the most significantbit (MSB) and the other the least significant bit (LSB), theaforementioned algorithm can obtain the impulse response ofthe MSB PRBS, the LSB PRBS, and the product of these twostreams. This information can be used to uniformly spread thePAM4 symbols and optimize the 3 eyes to almost equal eyeheight.V. EXPERIMENTAL SIMULATIONEAMEAM90°100 KmSMF 2890° OpticalHybride13 dBm7 dBmy1y2PRBS9 u(t-110Ts)PRBS9 u(t)Fig. 3: VPI simulation of coherent receiver, 100 Km fibercause self-phase modulation (SPM).Received Linear EQ Proposed NL-EQRotationIQ imbalanceReflectionNon-linear INon-linear IINon-linear PAM4TABLE I: Matlab simulation, to verify the proposed non-linearequalizer, for the most come channel distortion effect.The algorithm proposed in this paper can be used to describeand compensate various non-linear complex channels. To il-lustrate this in a more practical scenario, an optical experimentperformed in VPI is set up. The simulation consists of acoherent transmitter constructed with EAMs and depicted inFig.3. Self-phase modulation (SPM) of the signal through thefiber is one of the limited factors of the data throughput of acoherent receiver [3]. To simplify the transmitter, asymmetricQPSK symbols, namely {0,0},{0,1},{1,0},{1,1} are transmit-ted. The QPSK symbols are made of two PRBS9 test-patterns,the second has a delay of 110 symbols. The same methodis used, as described in Section IV. The PRBS test-patternsmodulate with an electro-absorption modulator (EAM) the In-phase and Quadrature term of a continuous wave (CW) laserlight. The non constant envelop of the transmitted signal causea phase fluctuation in fiber, resulting in SPM which is a formof AM-PM distortion. The fiber has a length of 100 Km and itsnon-linear index n2 = 2.6E-20 m2W , which is typical for SMF-28. The light entering the fiber has a power of 7 dBm and ismodulated with a symbol rate of 1 GBaud. A scatter plot ofthe received QPSK constellation is shown in Table II. A linearequalizer is not capable of placing the received symbols on theright coordinates. However, the proposed non-linear equalizercan completely compensate the non-linearity.Received Linear EQ Proposed NL-EQTABLE II: VPI simulation of coherent receiver, 100 Km fiber,from left to right: the received QPSK constellation, distortedby SPM, is shown, the linear equalized received constellationand the received signal after the proposed non-linear equalizer.VI. CONCLUSIONThis paper described an algorithm to equalize non-linear ef-fects in a MIMO system. First, it is shown that the polynomialmatrix describing a MIMO system can be extended to includenon-linear effects. Second, pseudo random bit sequences areused to estimate the different elements constituting this ex-tended polynomial matrix, and last, this estimated matrix isused to equalize data received through the non-linear channel.To show the validity of this approach, different impairedchannels are simulated in Matlab and compensated with botha linear zero-forcing equalizer as well as with the proposedequalizer. It was shown that for example saturation of anunbalanced QPSK transmission could be compensated withthe proposed equalizer, while this is impossible with a linearequalizer. Additional, a VPI simulation of a similar system isperformed. Again an unbalanced QPSK signal is transmitted,but this time the distortion occurs due to self phase modulation.The proposed equalizer can easily compensate this. As such,it is shown that the method reported can be adjust or extendto cover multiple non-linear effects.ACKNOWLEDGMENTPiet Demeester thanks the ERC for his advanced grant695495 ”ATTO: A new concept for ultra-high capacity wire-less networks”.REFERENCES[1] B. Lannoo, A. Dixit, D. Colle, J. Bauwelinck, B. Dhoedt, B. Jooris,I. Moerman, M. Pickavet, H. Rogier, P. Simoens, G. Torfs, D. V. Ginste,and P. Demeester, “Radio-over-fibre for ultra-small 5g cells,” in 2015 17thInternational Conference on Transparent Optical Networks (ICTON), Jul.2015, pp. 1–4.[2] B. Goebel, G. J. Foschini, G. Kramer, P. J. Winzer, and R.-J. Essiambre,“Capacity Limits of Optical Fiber Networks,” J. Lightw. Technol., vol. 28,no. 4, pp. 662–701, Feb. 2010.[3] T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima,and K. Kasahara, “A comparative study of dpsk and ook wdm trans-mission over transoceanic distances and their performance degradationsdue to nonlinear phase noise,” J. Lightw. Technol., vol. 21, no. 9, pp.1933–1943, Sep. 2003.[4] N. Benvenuto and G. Cherubini, Algorithms for communications systemsand their applications. J. Wiley, 2002.[5] D. Darsena, G. Gelli, and F. Verde, “Joint blind channel shortening andcompensation of transmitter i/q imbalances and cfos for uplink sc-ifdmasystems,” Physical Communication, vol. 11, no. Supplement C, pp. 25 –35, 2014, radio Access Beyond OFDM(A).[6] A. Ahrens and S. Lochmann, “Polynomial matrix algorithms for broad-band optical mimo systems,” in 2016 Advances in Wireless and OpticalCommunications (RTUWO), Nov. 2016, pp. 22–27.[7] S. Lesecq and A. Barraud, “A prbs with exactly zero correlation andits application,” in 2008 16th Mediterranean Conference on Control andAutomation, Jun. 2008, pp. 724–728.[8] Y. Lin, “‘shift and add’ property of m-sequences and its application tochannel characterisation of digital magnetic recording,” IEE Proc. Comm.,vol. 142, no. 3, pp. 135–140, Jun. 1995.

https://biblio.ugent.be/publication/8544367/file/8544369.pdf

Nonlinear MIMO impulse responses determination using pseudo random sequences, and nonlinear compensation

Abstract

Similar works

Full text

Available Versions

Crossref

Ghent University Academic Bibliography