Contains a report on a research project.National Science Foundation (Grant G-16526)National Institutes of Health (Grant MH-04737-03)National Aeronautics and Space Administration (Grant NsG-496

Schetzen, M.

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COMMUNICATION SCIENCESANDENGINEERINGXI. STATISTICAL COMMUNICATION THEORY*Prof. Y. W. LeeProf. A. G. BoseProf. M. SchetzenProf. H. L. Van TreesProf. V. R. AlgaziR. AlterA. B. BaggeroerR. F. BauerC. J. BoardmanJ. D. BruceM. BushChessmanD. CollinsCrimiA. DelagiA. FroeschleE. GlaveGreenblattG. KincaidC. KurtzD. E. NelsenA. V. OppenheimM. O. PaceR. B. ParenteM. A. RichP. M. RegoM. J. SchafferJ. E. ShindallD. W. SteeleC. I. M. UchoaA. SYSTEMS OF THE CLASS NONLINEAR NO-MEMORYWITH MEMORYFOLLOWED BY LINEARIn this report, we present some results concerning the class of nonlinear systemsof the form shown schematically in Fig. XI-1. As shown, the nonlinear system canbe divided into three sections that are connected in tandem: a nonlinear no-memorysystem whose transfer characteristics, Nm[x(t)], form a complete set of functions; aFig. XI- 1.multiple-input, multiple-output linear section whose impulse responses, h. (t), form acomplete matrix of linear operators; and a linear section consisting of amplifiers whoseoutputs are then summed. This class of nonlinear systems is not as general as thatdescribed by Wiener ; it is a subset that we shall describe in this report.We first observe that, by the use of superposition in linear systems, the systemof Fig. XI- 1 can be reduced to the form shown schematically in Fig. XI-Z. For thissystem, the set of impulse responses, hn(t), form a complete set. Thus the nonlinearsystems of the class that we are discussing are each of the form shown in Fig. XI-3, inwhich k (t) = a mnh (t). In order to analyze these systems, we shall approximatenThis work was supported in part by the National Science Foundation (Grant G-16526),the National Institutes of Health (Grant MH-04737-03), and in part by the National Aero-nautics and Space Administration (Grant NsG-496).QPR No. 73 131(XI. STATISTICAL COMMUNICATION THEORY)Fig. XI-2.the complete set of linear operators by the set of delayed impulses, u(t-tn). Such a setis obtained by taps on a delay line. Any linear system can be approximated arbitrarilyclosely by this set. That is, the set of delayed impulses is complete in the limit as thedelay between impulses approaches zero so that the set becomes dense on the t axis.N1  k1 (t)x(t z(t)Nm km(t)Fig. XI-3.The system of Fig. XI-2 can thus be approximated arbitrarily closely by the delayedimpulses as shown in Fig. XI-4. The output, z(t), of this system isz(t) = a Nm[x(t-t )],m nwhich is identical with the output of the nonlinear system shown in Fig. XI-5. Thusboth systems are equivalent.Each of the nonlinear systems shown in Fig. XI-5 are of the form shown in Fig. XI-6in which Mn[x(t)] = Z amnNm[x(t)]. Thus, as the delay between impulses approachesmzero, the class of systems of the form shown in Fig. XI-6 is equivalent to that ofthe form shown in Fig. XI-3. From Fig. XI-6, the nonlinear systems can be describedgenerally as those whose output is the sum of a different nonlinear no-memory functionof x(t) for each instant of time in the past; there are no cross products between the inputQPR No. 73 132(XI. STATISTICAL COMMUNICATION THEORY)Fig. XI-4.at different instants of time. Since the set of nonlinear no-memory operators, {Nm}, iscomplete, the system of Fig. XI-2 is complete in the sense that any set of nonlinear no-memory functions of x(t) for each instant of time in the past can be realized. In thisconnection we should note that the system of Fig. XI-6 is not equivalent to that ofN1 allx (t) z (t)N1 'alnu(t-tn) .Nm amnFig. XI-5.Fig. XI-7, in which the impulse responses, hn(t), are not delayed impulses. This iseasily seen, since the output of each of the nonlinear no-memory systems will containcross products between the input at different instants of time. The system of Fig. XI-7is a subset of those described by Wiener which contain no cross products between theu(t -t ) M1x (t)+ z(t)u(t -tn) MnFig. XI-6.l (t) Mx(t) . . z (t)hn (t) MFig. XI-7.QPR No. 73 133(XI. STATISTICAL COMMUNICATION THEORY)outputs of the linear systems. We observe from our discussion that the class of non-linear systems of the form shown in Fig. XI-7 is dependent upon the set of linear sys-tems used even though the set of impulse responses forms a complete set. Only if thecross products between the outputs of the linear systems are included, so that the classbecomes that described by Wiener, will it be independent of whatever complete set oflinear systems is used.M. SchetzenReferences1. N. Wiener, Nonlinear Problems in Random Theory (The Technology Press ofMassachusetts Institute of Technology, Cambridge, Mass., and John Wiley and Sons,Inc., New York, 1949).QPR No. 73 134

Statistical Communication Theory

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