{"title": "Blind Separation of Radio Signals in Fading Channels", "book": "Advances in Neural Information Processing Systems", "page_first": 756, "page_last": 762, "abstract": "", "full_text": "Blind Separation of Radio Signals \n\nFading Channels \n\n\u2022 In \n\nKari Torkkola \n\nMotorola, Phoenix Corporate Research Labs, \n\n2100 E. Elliot Rd, MD EL508, Tempe, AZ 85284, USA \n\nemail: A540AA(Qemail.mot.com \n\nAbstract \n\nWe apply information maximization / maximum likelihood blind \nsource separation [2, 6) to complex valued signals mixed with com(cid:173)\nplex valued nonstationary matrices. This case arises in radio com(cid:173)\nmunications with baseband signals. We incorporate known source \nsignal distributions in the adaptation, thus making the algorithms \nless \"blind\". This results in drastic reduction of the amount of data \nneeded for successful convergence. Adaptation to rapidly changing \nsignal mixing conditions, such as to fading in mobile communica(cid:173)\ntions, becomes now feasible as demonstrated by simulations. \n\nIntroduction \n\n1 \nIn SDMA (spatial division multiple access) the purpose is to separate radio signals \nof interfering users (either intentional or accidental) from each others on the basis \nof the spatial characteristics of the signals using smart antennas, array processing, \nand beamforming [5, 8). Supervised methods typically use a variant of LMS (least \nmean squares), either gradient based, or algebraic, to adapt the coefficients that \ndescribe the channels or their inverses. This is usually a robust way of estimating \nthe channel but a part of the signal is wasted as predetermined training data, and \nthe methods might not be fast enough for rapidly varying fading channels. \nUnsupervised methods either rely on information about the antenna array manifold, \nor properties of the signals. Former approaches might require calibrated antenna \narrays or special array geometries. Less restrictive methods use signal properties \nonly, such as constant modulus, finite alphabet, spectral self-coherence, or cyclo(cid:173)\nstationarity. Blind source separation (BSS) techniques typically rely only on source \nsignal independence and non-Gaussianity assumptions. \n\nOur aim is to separate simultaneous radio signals occupying the same frequency \nband, more specifically, radio signals that carry digital information. Since linear \nmixtures of antenna signals end up being linear mixtures of (complex) baseband \nsignals due to the linearity of the downconversion process, we will apply BSS at \nthe baseband stage of the receiver. The main contribution of this paper is to \nshow that by making better use of the known signal properties, it is possible to \ndevise algorithms that adapt much faster than algorithms that rely only on weak \nassumptions, such as source signal independence. \n\nWe will first discuss how the probability density functions (pdf) of baseband DPSK \nsignals could be modelled in' a way that can efficiently be used in blind separation \nalgorithms. We will incorporate those models into information maximization and \n\n\fBlind Separation of Radio Signals in Fading Channels \n\n757 \n\ninto maximum likelihood approaches [2, 6). We will then continue with the maxi(cid:173)\nmum likelihood approach and other modulation techniques, such as QAM. Finally, \nwe will show in simulations, how this approach results in an adaptation process that \nis fast enough for fading channels. \n\n2 Models of baseband signal distributions \nIn digital communications the binary (or n-ary) information is transmitted as dis(cid:173)\ncrete combinations of the amplitude and/or the phase of the carrier signal. After \ndownconversion to baseband the instantaneous amplitude of the carrier can be ob(cid:173)\nserved as the length of a complex valued sample of the baseband signal, and the \nphase of the carrier is discernible as the phase angle of the same sample. Possible \ncombinations that depend on the modulation method employed, are called sym(cid:173)\nbol constellations. N-QAM (quadrature amplitude modulation) utilizes both the \namplitude and the phase, whereby the baseband signals can only take one of N \npossible locations on a grid on the complex plane. In N-PSK (phase shift keying) \nthe amplitude of the baseband signal stays constant, but the phase can take any \nof N discrete values. In DPSK (differential phase shift keying) the information is \nencoded as the difference between phases of two consecutive transmitted symbols. \nThe phase can thus take any value, and since the amplitude remains constant, the \nbaseband signal distribution is a circle on the complex plane. \nInformation maximization BSS requires a nonlinear function that models the cu(cid:173)\nmulative density function (cdf) of the data. This function and its derivative need \nto be differentiable. In the case of a circular complex distribution with uniformly \ndistributed phase, there is only one important direction of deviation, the radial \ndirection. A smooth cdf G for a circular distribution at the unit circle can be \nconstructed using the hyperbolic tangent function as \n\nG(z) = tanh(w(lzl - 1)) \n\n(1) \n\nand the pdf, differentiated in the radial direction, that is, with respect to Izl is \n\n8 \n\ng(z) = Bizi tanh(w(lzl - 1)) = w(l - tanh2(w(lzl - 1))) \n\n(2) \nwhere z = x + iy is a complex valued variable, and the parameter w controls the \nsteepness of the slope of the tanh function. Note that this is in contrast to more \ncommonly used coordinate axis directions to differentiate and to integrate to get \nthe pdf from the cdf and vice versa. These functions are plotted in Fig. 1. \n\na) CDF \n\nb) PDF \n\nFigure 1: Radial tanh with w=2.0 (equations 1 and 2) . \n\nNote that we have not been worrying about the pdf integrating to unity. Thus we \ncould leave the first multiplicative constant w out of the definition of g. Scaling will \nnot be important for our purposes of using these functions as the nonlinearities in \nthe information maximization BSS. Note also that when the steepness w approaches \ninfinity, the densities approach the ideal density of a DPSK source, the unit circle. \nMany other equally good choices are possible where the ideal density is reached as \na limit of a parameter value. For example, the radial section of the circular \"ridge\" \nof the pdf could be a Gaussian. \n\n\f758 \n\nK. Torkkola \n\n3 The information maximization adaptation equation \nThe information maximization adaptation equation to learn the unmixing matrix \nW using the natural gradient is [2] \n\nAWex (guT + I)W where \n\n(3) \nVector U = W x denotes a time sample of the separated sources, x denotes the \ncorresponding time sample of the observed mixtures, and Yj is the nonlinear function \napproximating the cdf of the data, which is applied to each component of the u. \nNow we can insert (1) into Yj. Making use of {)lzI/{)z = zllzl this yields for 'OJ: \n\n!!Jli.. \n8 \n8Yi 8Ui \n\n~ . -\nYJ -\n\n~ \nYj = -() -() tanh(w Uj -1 = -2WYj-1 -I \nUj \nUj \n\n() \n() \nYj Uj \n\n(I I ) ) \n\nWhen (4) is inserted into (3) we get \n\n