The Kolmogorov mapping theorem in signal processing

THE KOLMOGOROV MAPPING THEOREM IN SIGNAL PROCESSING Miguel A, Lagunas ETSl de Telecomunicacion, Departamento de TSC Apdo. 30002, 08080 ABSTRACT' Since the publication in 1957 of the work of Andrei Kolmogorov 181 in mapping a function of multiple variables by means functions of a single variable, many mathematicians and engineers try , with different degree of success and not without controversy 1191. to find the direct application of it to multiple extremes problems, rooting of multivariate polynomials, neural networks and pattern recognition. This paper revisits the theorem from the optic of a generalised architecture for signal processing 1281. It is envisaged the high potential of the theorem to handle either linear or non-linear processing problems. A specific implementation following the main guide-lines of the theorem is reported, as well as some preliminary results concerning the design, implementation and performance of non-linear systems. The applications cover non linear transmission channels for communications, instantaneous companders and prediction of chaotic series. I. INTRODUCTION Regardless linear systems have been the main component of communications and signal processing engineering, with a clean and forinal mathematical background, it is interesting lo note the massive presence of non-linear systems and devices in modern communicatio~~s equipment. Analog mixers and modulators, companders, synchronism units, decoders are examples of a vast list from which it can be said that the role of non-linear systems components and devices is. most of the cases, ahsolutely crucial in the global systems when compared with the corresponding linear part of them. Also. thc inherent nonlinear hehaviour of many amplifiers and filters. when handling high dynamic ranges, put in sccne the issue of linearisation in the analog domain. Surprisingly. there is not a formal manner to modcl a non-linear system neither an architecture with support the reportcd schemes with a broad ambit of application, heing most of the works reported dealing with the prol~lcni of gaussiion input /14//15//16//25/26/127/. 111 fact. it can hc said that polynomial modclling constitutes the only way to work with non-lincar filters. At the s:iinc time. cascading non- BARCELONA, SPAIN linear memory less with convolutive systems, using the two possible alternatives, was considered as a serious attempt to implement non-linear equalisers for dispersive communication channels in the seventies. More recently, two topics emerged in an almost separate manner. Neural networks 1'21 contributed, at the architecture level, to increase the interest and viability of non-linear processing schemes. On the other hand, high order processing 13//4/, at the algorithmic level, were in the scene as a basic tool to increase the scope of, let us say, classical processing system developed from second order functions. In any case, th: architecture and the processing tools efforts seem to go ahead one apart from the other, without an uniform and global approach to the non-linear processing framework. The purpose of this work was, at the beginning three years ago, to find out refreshing mathematics, in the line of what has been mentioned in the previous paragraph, focussing the linearisation problem of communication components. The theoiem reported by Kolmogorov concerning the representation of multivariate functions with single variable functions 1201 shown the link, in some sense especulative, between neural networks, high order processing and the EM aigorithm /6//5/. In fact, a lateral inhibition network with high order learning is a clarifyed example concerning the implementation of the EM algorihtm's principle. The application reported was a DOA tracker for moving sources which, in a blind context, generates a dedicated beam for every source impinging the aperture, for radial velocities below one degree per snapshot. No matter the mapping theorem was the basic motivation to encompass .he thrce techniques in a source separation prohlem 17/.9/11011111112/113/. no specific contribution to the architectural design of non-linear systems was concluded fiom this work. More interesting than the hlind DOA tracker was a system, with the same background. designed to solve collision problems in a time reference heamforiner for inohile communications. The architecture was again a latcral inhibition network, with two cxtcnded Kalman filters to acquire and track two siinultancous carriers. which act as a call signal dcmanding a dedicatcd beam. The lcjrning was high order for the iii hi b i tio n pen a 1 ty in acquisition. convergence mte. hcnmforining arid complexity with respect the singlc call system /30//31/. This last systcrn was an examplc of up to what degree the EM ;ilgorithni involves an archilccture. heing the scope of we i g h ts , w I ! ho u t subs tan t i a1 - 185 - applications enlarged when the estimation steep is considered as an inhibibon one, which reduces the multlple extrema problem to a single maximum objective. Coming back to the mapping theorem, it was considered the importance of the architectural aspect involved in the Kolmogorov's theorem. In fact, what is going to be reported hereafter is more an architecture than an algorithm for signal processing. Ir seems clear that most of the possibilities of success in non-linear processors, as well as in neural network approach for a problem solver, are based in the architecture choice, locating in a secondary plane the corresponding algorithm for design andlor learning. The architecture will be refereed from now on as the kol-filter. Regardless the paper focuss its content in nonlinear filtering, the impact of the theorem in many different problems, with different applications, rises the doubt of up to what degree the reported scheme is just a very small part of the potential applications and consequences of it. The author is highly convinced that the theorem will be still alive and that the reported material are really very small and specific results, with a high component of speculation and lack of formality, that in any case do not preclude further contributions from the work of Andrei Kolmogorov. 2. THE MAPPING THEOREM. The mapping theorem due to A. Kohogorov proves the existence of a two stage architecture to map and input vector in an output vector y. Being an existence theorem, there is not reported yet any practical implementation, for non-linear processing, following in strict senke its guidelines. In fact, it seems that the issue will remain in the current status and that, in future work, the theorem will act as fundamental motivation to formalise the solutions of application driven problems. It may be that this way is the adequate manner to arise at a full profit, at the engineering level, of a mathematical work of outstanding interest in signal processing. No matter in the literature we can find, together with the original version, many descriptions as well as fundamentals and controversy about the theorein, hereafter we will reduce its reading tu the aspects of interest for signal processing; further reading in the theorein can be found in the following references /8//17//19//18//20/. Given a continuous mapping E(.) function of 11 input 8variahles, the components of the input vector &, it exists an architecture, based in single variable functions that implements the mapping ~-->g(&). The architecture contains two stages of processing. Selecting the case of siuglc output (i.e. the output vector reduces to a scalar y). the theorein in its original formulation had the following representation: 2n n Later on, it u'as rcportcd rhc possibility of changing functions y by a singlc f'uiictivn y ( without any change in the definition interval): at the saint time. fuiictions @( .) of P the first stage can be replaced by 2n+l functions +q (q=1.2n+l) plus a chosen parameters X which depend on the number of input componznts n. The formulation that is going to be used here is another possibility, which reduces to a single function the first stage and preserves the 2n+l of the second stage. Thus, the first stage of processing is formed by (2). P n where the function or transform represented by @(.) does not depend on the specific mapping, and the parameters h and E also do not depend on the mapping, but they do on the number of components of the input vector n. q q The second stage of processing is formed by 2n+l (.), which, by direct superposition, form the functions y output function y. P 2nt 1 Note that the most interesting part of the theorem is the, let us say, absolute value or the wide amhit of application of the functions forming the fist stage. It seems to he like a general trans:orm which is mandatory t3 any processor architecture for the mapping problem. An approach that keeps some similarity with this transform is the use of saturated de* ices (limiters) prior a Volterra approximation in model!ing a compander. This scheme arises when a given non-linearity is modelled by a polynomial approximation. The approximation by itself does not include saturation , in consequence, and to avoid that the polynomial approximation of the compander may produce out of range values at the output, it is necessary to saturate the dynamic of the input values applied to the Volterra system. Fig. I. A transform before the Volterra approximation of a givcn compander helps to avoid high dynamic output ranges and protects later stage:. in communications equipment. At the same time. t IS well known the extraordinary pcrlorinance of logarithm coinpaodcrs in many applications lor coding and noise red iction both in speech and high ficlclity audio /23//24/ Somehow the mentioned experience ul~out ccrtain traiihforin\. together with what 1s stated in the - 186 - theorem, for the first stage of processing, arise to the conclusion that it may exist a data transform which. independently of the mapping and connected with the extended use of logarithmic and sigmoidal companders, reduces the complexity and improves the performance of a second stage of processing based only in Volterra systems. As we will see later on this paper, this architecture has a formal approach and somehow provides additional reasoning to the use of the above functions. To be more specific, the architecture of the kol-filter will be a first stage formed by phase modulators (PM) followed by Volterra filters. Before we pass to the next section. note that the problem to be considered is the filtering one, where a data diversity input, that could he either time ( FIR filters) or spatial (array processing), is the input to the processor which will map this data diversity in a given output function YO). y(t)=g(x(t),x(t-T), ..... x(t-Q.T)) (4) 3.VOLTERRA, FOURIER AND KOL-FILTERS. Regardless the main interest of non-linear processing use to be reduced to the case of data diversity input mapped to a single output (i.e. non-linear system with memory), the presentation will start with the case of non diversity data input. When adequate, an indication will be provided about the extension to the general case. Thus, assuming a memory less non-linear function g(.). Y (t)=s(x(t)) (5 ) the polynomial approximation models the ahove system as (6), which will he refereed as the Volterra filter or model. Q q=1,3,.. yv(t)= c d(q) .x% (6) The corresponding filter for the case of diversity in the input reduces to include kernels with all suitable combinations of the diversity components, both in number and exponents. Continuing with the diversity case. a reduced complexity schemes, at the expense of approximation quality of the actual mapping, are the so-called Wiener and Hammerstein architectures /1//22/. which basically are the cascade of P non-linear system without meinory with a linear system or viceversa. An alternative to the above representation, more in line with the architecture involved in the mapping theorem, is to use a Fourier representation of the inapping. as it is indicated in (7). m yf(t)= J G(w) , exp[j.w.x(t)) . dw (7) -00 Furthermore, given the input dynainics. the ahove mapping can be reduced to a Fouricr Series which, under a11 approximation error. cui hc rcducctl ((1 il fiiiitc Fuuricr SCII'IC~ as in (8). Before going irto details on the mentioned approximation, note that the Fourier model keeps a close relationship with the theorem, in the sense that a frst stage of PM modulation is always required, independently of the mapping itself, and a second stage, specific for the given mapping function , follows. This second stage can be implemented with a Volterra filter. [ exp(j.wo.x(t)) WO Fig 2. Fourier model of the. mapping function showing some similarity with the two stage processor of the mapping iheorem. Some other guide-lines of the theorem are pending like the necessary 2Q+1 functions of the first stage and the parameters h and E. In the case that the mapping has an odd symmetry the above model can be represented as a power series of noncomplex PM modulators. At the same time, to do not exclude the case of non-cero output for Zero input, the dc term has to be added to the model as j: is denoted in (9). where the sine model has been reformulated in tenns of the hannonics of the PM carrier frequency. 1 QI yf(t)=h(O)+ b(q).sinq(w,.x(t)) (9.a) q= 1 Q q= 1 yf(t)=c(O)+ C c(q).sin(q.wo.x(t)) (9.b) With respect the dc term in the above formulas, note that, for the case of (9.a), it will .le necessary, even there is a zero input to zero output ccuversion in the mapping, since ititcinal modulation produt LS may produce a dc term that has to be adjusted or ccimpensated to reproduce the corresponding mapping fu,iction. At this moment, it can he said that for the case of Q-components input it is necessary to have II I'M modulators plus the dc tenn. In other words. it relnains Q additional functions to match the above processor with the theorem. Before going ahead. it is worthwhile to compare the approximations associatcc, to the single Volterra model and the Fourier model. Both! approximations are in principle equivalent since. assui->inp no over modulation (i.e. Q.w~,.x(I)<~cK), the exponential term in (7) can be - 1137 - approximated by a Taylor's series producing very generic form, relates the two approaches 10) which, in a 90 Nevertheless, the above derivation reduces to a formal relationship since the convergence of the two approximations, with increasing order, to the actual mapping could be quite different ( an extreme case is for sinusoidal-like mappings which are very common in practice). More important that the previous argument, in an adaptive design of the approximation weights. the bounded dynamic range of the Fourier kernels acts very much in favour of it, with respect the Volterra modelling, improving dramatically the convergence rate for similar adaptive algorithm or learning rule. Also, it should be noticed that the cross-correlation of the kernels involved in each approximation, easy to compute for the case of gaussian distributed input data, is lower in the Fourier model than in the Volterra case; this improves convergence rate and provides an effective order reduction for similar levels of performance. With respect to the design of the CO-efficients, it can be carried out by direct frequency sampling of the Fourier transform of the mapping G(w), when the desired mapping is a priori known, or by a LMS like algorithm for the sinusoidal kernels, using standard MSE objectives. The extension of the model to the case of data diversity input is done by setting a PM modulator for every diversity component followed by a specific Volterra system for every branch. This generates n+l functions z (t); the addition of these functions provides the output of the filter. P z (t)= sin (w . x(t-pT)), p=l,n (11) P P f0 n (12.b) y(t)= c fp(t) (13) p=o This structure makes evident the coinbination of PM modulators, with different frequency for every diversity component, and the specific Voltcrra systems. Let us to revisit. at this 1noinent. the approximation involved in passing from the continuous Fourier model to the Fourier series in (8). Note that the formulation we arise in (11)-(13) sets up the roll of the I'M modulators. followed by memory less Voltena systems. as the way out to implement the sampling of the continuous Fourier Model. In other words, (1 1) with (12) represent the multidiniensiona1 sampling of the corresponding Fourier representation of the model. This sampling is uniform and concentrated in the axes of the multidiniensional function G(wl ,.... wp ..... w ), but not in the planc. which should he more adequate in general. In this sampling prolilem are iinl~cded two arguments claimed in thc inappinp thcurcm as wc will sec in the next paragraphs. Q About the sampling involved in the Fourier series representation of the contiiiuous model, and focussing again the case of nondiversity input, the corresponding model can be re-formulated as( 14). zl(t)= sin (wl. x(t)) (14.a) 0 f0 1 (14.c) (14.d) where it is evident the uniform sampling performed by the Volterra system at the harmonics of the modulation frequency. Related with the problem of sampling, it is interesting to remind the important consequences of random sampling for discrete processing in order te emulate continuous systems. It is :>ossible by suitable randomising schemes to design, and to implement, discrete systems with reduced, in practice negligible, aliasing effect (see 1211 for a detailed study on randomised signal processing). The basic idea is to achieve an almost uniform distribution of sampling intervals because the expected value of the discrete processing converges to the actual or continuous value without aliasing effects.In consequence, a better approach would be to randomise the mentioned sampling in order to obtain an approximation y kol(t), such that the expected value of it coincides with the actual function y(t). Just to describe briefly the effect.: claimed under the randomised assumption, and taking .us as the random samples of the continuous G(w), the expected value of the model output will he: where, it is clear that, whenever the distributions of the sampling frequencies ove.lap in a constant or uniform pdf, the expected value will coincide with the actual one. One way out to produce an almost random sampling. or scanning. of the frequency axis is to modify the Volterra system that follows the PM modulator; nevertheless. this will increase the iinpleinentation complexity and somehow will deviate thc architecture from the theorcm. in the sense that the random samplinp, effect takes part inside the first stage of processing, i.e. it should he independent of the mapping itself. The inann-r selected to produce this random sampling effect is to a:.d another I'M modulator with modulation frequency slightly deviated from the initial one. This implies the presence of another I'M modulator with modulation frequency equal to w + E Note that this ;itlditional modulator inay introduce the random effect whcncver the harmonics of the second frequency do not - 188 - coincide with the original one. In fact, taking an MSE technique to find out the co-efficients. c(q) in (14), the selection of the perturbation influences the condition number of the corresponding kernel matrix. Note also that in the case of diversity data input. the selection of the perturbations E (p=l,Q) depends on the number of input diversity components Q and not in the specific mapping, as it was anticipated in the existence theorem. At the same time, the perturbed modulators, together with the dc term, seems to be the necessary 2Q+1 functions claimed in the theorem. P """""",,------------, _-. I I r I' 1 I' I C,r'----------- - I! ------------ I Figure 3. The architecture of the Kol-filter. The three steeps are imbeded in dashed syuares, these squares indicate the diversity stage, the transforms or phase modulators with the nominal and perturbed frequency plus the linear combiner. and the specific Volterra systems. Still there is a prohlein associated with sampling only along the freyuency axis and not in the multidimensional frequency plane. In other words, with the current scheme. every diversity component is scanned, almost at random, in its coi~esponding frcqucncy axis of the function G(wl ..... w .... w,,). It is clear that combining [he modulated versions prior to the Volterra processing will produce a frequency scan in the ~nultidimc~~sional frequency plane. leaving to the Vollcrra order the COIIVC~~CIIC~ lo the actual system to model. The solution was [(I inclutlc a linear combiner of the ~nodula~ctl diverbily compoilents. prior to P' the Volterra systems of the second stage, which was formulated as (16). 1 oo., 0 110..0 c=111..0 111..: This combiner, which is also independent of the specific mapping. seems to play the role assigned to parameters h included in the theorem ( see (2)). It is clear that the choice for the linear combiner reduces to a very specific scanning mode of the multidimensional frequency plane. In the case that some a priori knowledge of the mapping Fourier transform is available, both, the selection of modulation frequencies, pcrturbations and combiner, could be oriented to ensure aJequate sampling of every local extremes of the mentioned Fourier representation of the mapping. The architecture that summarises all the points included in this section, refereed herein as the kol-filter, is depicted in figure 3. It is worthwhile to remind that this structure includes a dedicated processing foi every diversity branch, being a steep ahead from the coarse approximation which is associated with the existing alternatives, refereed as Wiener and Hammerstein structures. Finally, note that the Volterra kernels can be designed with and MSE criteria, when a training output is available or by direct sampling of the multidimensional Fourier function associated with the system. The adaptive versions can be applied also to the mentioned co-efficients, having the benefit of the bounded dynamic range of the corresponding kernels. 4. SOME APPLICATIONS OF INTEREST The kol-filter was ked to equalise a communication system with non-linear hc:!iaviour in the transmitter (above 6% distortion to the secoi:d and third harmonic), dispersive communication channel, ai;d non-linear receiver effects (2% distortion to the second harmonic). The second stage weights were estimated by using a MSE objective. The performance of the kol-filter was highly above the corresponding Volterra or Hammerstein alternatives. The residual, with respect the undistorted transmitted signal was 2% of the reference power. 16 and 42% for the kol-filter. Volterra and Hammerstei 1 respectively. A very important result was that even when :io distortion was set in the overall coinmunication link (i.e. the system to he equalised was linear) the kol-filter shown the same performance that the optimum linear Wiener filt,?r. Other aspect of interest is the ability of the herein reported filter to predict, with no appreciable error. chaotic signal series. The so-citlled Tent-map was used to evaluate thc prediction capabilities of the non-linear kol-filter with excellent results. The design. block or adaptive. of instantaneous cornpanders was done in :.xnparison with classical Volterra models. Again the yua!ity of the mapping. in block processing mode. and also the convergence rate in adaptive - 189 mode were dramatically good when compared with the classical alternative. Similar results were obtained for linearising power amplifiers. reducing the scheme to an array of medium power amp!ifiers with no relevant distortion in the global response. Finally, the kol-filter ideas has been used in order to obtain high rate 1-bit AD and D/A converters. Being this work at a very preliminary stage, the result are very promising to encourage further work in the topic as a valid alternative to the CD converters used currently. 5. CONCLUSIONS The purpose of this work was to shown the potential associated to an important and, at the same time, more surprising theorem concerning non-linear mapping or filtering. Motivated and written by an engineer, the style of the paper has been speculative and positive, versus the relevance of the theorem and the extraordinary contribution of Andrei Kolmogorov. Probably, almost certain, that this work can be improved even in fundamental aspects of it. The search for the unknown transform of the first stage. even the special character of it, seems to be the main issue to come out with any efficient implementation of the theorem. 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